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Single Carrier Orthogonal Multiple Access Technique for Broadband Wireless Communications DISSERTATION Submitted in Partial Fulfillment Of the Requirements for the Degree of DOCTOR OF PHILOSOPHY (Electrical Engineering) at the POLYTECHNIC UNIVERSITY by Hyung G. Myung January 2007

Approved:

___________________ Department Head Copy No. _____________

________________________ Date

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Copyright by Hyung G. Myung 2007

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Approved by the Guidance Committee: Major: Electrical Engineering

______________________________ David J. Goodman, Ph.D Professor of Electrical and Computer Engineering

______________________________ Peter Voltz, Ph.D Associate Professor of Electrical and Computer Engineering

______________________________ Elza Erkip, Ph.D Associate Professor of Electrical and Computer Engineering

______________________________ Donald Grieco Senior Manager of InterDigital Communications Corporation

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Microfilm or other copies of this dissertation are obtainable from: UMI Dissertation Publishing Bell & Howell Information and Learning 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Michigan 48106-1346

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Curriculum Vitae Hyung G. Myung received the B.S. and M.S. degrees in electronics engineering from Seoul National University, South Korea in 1994 and in 1996, respectively, and the M.S. degree in applied mathematics from Santa Clara University, California in 2002. He received his Ph.D. degree from the Electrical and Computer Engineering Department of Polytechnic University, Brooklyn, NY in January of 2007. From 1996 to 1999, he served in the Republic of Korea Air Force as a lieutenant officer, and from 1997 to 1999, he was with Department of Electronics Engineering at Republic of Korea Air Force Academy as an academic instructor. From 2001 to 2003, he was with ArrayComm, San Jose, CA as a software engineer. During the summer of 2005, he was an assistant research staff at Communication & Networking Lab of Samsung Advanced Institute of Technology. Also from February to August of 2006, he was an intern at Air Interface Group of InterDigital Communications Corporation, Melville, NY. Since January of 2007, he is with Qualcomm/Flarion Technologies, Bedminster, NJ as a senior engineer. His research interests include DSP for communications and wireless communications.

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To Christ my savior, Hyun Joo, and Ho

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Acknowledgements During the past three years working towards my PhD degree, I was very fortunate enough to come across many great individuals and I am very grateful for it. I would like to give the utmost gratitude to my thesis advisor, professor David J. Goodman. Not only was he generous enough to guide my thesis research during his busy schedules, but he was also my role model as a great engineer and teacher. Through numerous discussions and one-on-one meetings, I learned so much from him and I greatly appreciate all the advice and wisdom, big and small. I would like to thank the members of the guidance committee, professor Peter Voltz, professor Elza Erkip, and Donald Grieco, for their time and valuable feedback on my research. I am honored to have them on the committee. I also wish to express my special appreciation to Dr. Junsung Lim and Kyungjin Oh with whom I carried out joint research on SC-FDMA resource scheduling. I thank them for the many hours spent together doing research and encouraging each other. I am also grateful to my parents for their support and encouragement to pursue the PhD study. Lastly, I would like to thank my wife and soul mate Hyun Joo who stood behind me rain or shine. Her loving and caring words were sources of encouragement to me and I am deeply grateful for them.

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An Abstract Single Carrier Orthogonal Multiple Access Technique for Broadband Wireless Communications by Hyung G. Myung Advisor: David J. Goodman

Submitted in Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy January 2007

Broadband wireless mobile communications suffer from multipath frequency-selective fading. Orthogonal frequency division multiplexing (OFDM) and orthogonal frequency division multiple access (OFDMA), which are multicarrier communication techniques, have become widely accepted primarily because of its robustness against frequency selective fading channels. Despite the many advantages, OFDM and OFDMA suffer a number of drawbacks; high peakto-average power ratio (PAPR), a need for an adaptive or coded scheme to overcome spectral nulls in the channel, and high sensitivity to frequency offset.

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Single carrier frequency division multiple access (SC-FDMA) which utilizes single carrier modulation at the transmitter and frequency domain equalization at the receiver is a technique that has similar performance and essentially the same overall structure as those of an OFDMA system. One prominent advantage over OFDMA is that the SC-FDMA signal has lower PAPR. SC-FDMA has drawn great attention as an attractive alternative to OFDMA, especially in the uplink communications where lower PAPR greatly benefits the mobile terminal in terms of transmit power efficiency and manufacturing cost. SC-FDMA is currently a working assumption for the uplink multiple access scheme in 3rd Generation Partnership Project Long Term Evolution (3GPP LTE). In this thesis, we first give a detailed overview of an SC-FDMA system. We then analyze analytically and numerically the peak power characteristics and propose a peak power reduction method that uses symbol amplitude clipping technique. We show that subcarrier mapping scheme and pulse shaping are significant factors that affect the peak power characteristics and that symbol amplitude clipping method is an effective way to reduce the peak power without compromising the link performance. We investigate multiple input multiple output (MIMO) spatial multiplexing technique in an SC-FDMA system using unitary precoded transmit eigenbeamforming with practical limitations. We also investigate channel-dependent scheduling for an uplink SC-FDMA system taking into account the imperfect channel information in the form of feedback delay. To accommodate both low and high mobility users simultaneously, we propose a hybrid subcarrier mapping method using orthogonal code spreading on top of SC-FDMA and show that it can have higher capacity gain than that of conventional subcarrier mapping scheme.

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List of Contents Curriculum Vitae

v

Acknowledgements

vii

An Abstract

viii

List of Figures

xii

List of Tables

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Chapter 1 Introduction

1

1.1.

Evolution of Cellular Wireless Communications.................................................................................1

1.2.

3GPP Long Term Evolution ...................................................................................................................2

1.3.

Single Carrier FDMA................................................................................................................................5

1.4.

Objectives and Contributions..................................................................................................................6

1.5.

Organization...............................................................................................................................................8

1.6.

Nomenclature...........................................................................................................................................10

Chapter 2 Channel Characteristics and Frequency Multiplexing

13

2.1.

Characteristics of Wireless Mobile Communications Channel........................................................13

2.2.

Orthogonal Frequency Division Multiplexing (OFDM) ..................................................................17

2.3.

Single Carrier with Frequency Domain Equalization (SC/FDE)....................................................19

2.4.

Summary and Conclusions.....................................................................................................................22

Chapter 3 Single Carrier FDMA

23

3.1.

Overview of SC-FDMA System...........................................................................................................25

3.2.

Subcarrier Mapping.................................................................................................................................28

3.3.

Time Domain Representation of SC-FDMA Signals........................................................................31

3.4.

SC-FDMA and OFDMA .......................................................................................................................36

xi 3.5.

SC-FDMA and DS-CDMA/FDE........................................................................................................38

3.6.

SC-FDMA Implementation in 3GPP LTE Uplink............................................................................40

3.7.

Summary and Conclusions.....................................................................................................................45

Chapter 4 MIMO SC-FDMA

47

4.1.

Spatial Diversity and Spatial Multiplexing in MIMO Systems .........................................................48

4.2.

MIMO Channel .......................................................................................................................................49

4.3.

SC-FDMA Transmit Eigen-Beamforming with Unitary Precoding ...............................................53

4.4.

Summary and Conclusions.....................................................................................................................60

Chapter 5 Peak Power Characteristics of an SC-FDMA Signal: Analytical Analysis

63

5.1.

Upper Bound for IFDMA with Pulse Shaping ..................................................................................64

5.2.

Modified Upper Bound for LFDMA and DFDMA..........................................................................70

5.3.

Comparison with OFDM.......................................................................................................................71

5.4.

Summary and Conclusion ......................................................................................................................72

Chapter 6 Peak Power Characteristics of an SC-FDMA Signal: Numerical Analysis

74

6.1.

PAPR of Single Antenna Transmission Signals..................................................................................76

6.2.

PAPR of Multiple Antenna Transmission Signals .............................................................................80

6.3.

Peak Power Reduction by Symbol Amplitude Clipping ....................................................................83

6.4.

Summary and Conclusions.....................................................................................................................88

Chapter 7 Channel-Dependent Scheduling of Uplink SC-FDMA Systems

90

7.1.

Channel-Dependent Scheduling in an Uplink SC-FDMA System..................................................91

7.2.

Impact of Imperfect Channel State Information on CDS...............................................................96

7.3.

Hybrid Subcarrier Mapping ................................................................................................................ 106

7.4.

Summary and Conclusions.................................................................................................................. 110

Chapter 8 Conclusions

111

Appendix A Derivations of the Upper Bounds in Chapter 5

115

Bibliography

126

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List of Figures Figure 2.1: Delay profile and frequency response of 3GPP 6-tap typical urban (TU6) Rayleigh fading channel in 5 MHz band. ..................................................................................................... 14 Figure 2.2: Time variation of 3GPP TU6 Rayleigh fading channel in 5 MHz band with 2GHz carrier frequency............................................................................................................................... 16 Figure 2.3: Transmitter and receiver structures of SC/FDE and OFDM. ..................................... 19 Figure 2.4: Dissimilarities between OFDM and SC/FDE................................................................. 21 Figure 3.1: Transmitter and receiver structure of SC-FDMA and OFDMA systems. .................. 24 Figure 3.2: Raised-cosine filter. .............................................................................................................. 27 Figure 3.3: Generation of SC-FDMA transmit symbols.................................................................... 28 Figure 3.4: Subcarrier mapping modes; distributed and localized. ................................................... 29 Figure 3.5: An example of different subcarrier mapping schemes for N = 4, Q = 3 and M = 12. ............................................................................................................................................................ 30 Figure 3.6: Subcarrier allocation methods for multiple users (3 users, 12 subcarriers, and 4 subcarriers allocated per user)........................................................................................................ 30 Figure 3.7: Time symbols of different subcarrier mapping schemes. .............................................. 35 Figure 3.8: Amplitude of SC-FDMA signals........................................................................................ 35 Figure 3.9: Dissimilarities between OFDMA and SC-FDMA........................................................... 37

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Figure 3.10: DS-CDMA with FDE. ...................................................................................................... 38 Figure 3.11: Spreading with the roles of data sequence and signature sequence exchanged for spreading signature of {1, 1, 1, 1} with a data block size of 4................................................. 39 Figure 3.12: Basic sub-frame structure in the time domain. .............................................................. 40 Figure 3.13: Physical mapping of a block in RF frequency domain (fc: carrier center frequency). ............................................................................................................................................................ 41 Figure 3.14: Generation of a block. ...................................................................................................... 41 Figure 3.15: FDM and CDM pilots for three simultaneous users with 12 total subcarriers. ........ 45 Figure 4.1: Description of a MIMO channel with Nt transmit antennas and Nr receive antennas. ............................................................................................................................................................ 50 Figure 4.2: Block diagram of a spatial multiplexing MIMO SC-FDMA system. ........................... 53 Figure 4.3: Simpified block diagram of a unitary precoded TxBF SC-FDMA MIMO system.... 54 Figure 4.4: Input-output characteristics of the quantizers. ................................................................ 57 Figure 4.5: FER performance of a 2x2 SC-FDMA unitary precoded TxBF system with feedback averaging and quantization. ............................................................................................................ 58 Figure 4.6: FER performance................................................................................................................. 59 Figure 4.7: FER performance of a 2x2 SC-FDMA unitary precoded TxBF system with feedback delays of 2, 4, and 6 TTI’s. ............................................................................................................. 61 Figure 5.1: CCDF of instantaneous power for IFDMA with BPSK modulation and different values of roll-off factor α. ............................................................................................................. 68 Figure 5.2: CCDF of instantaneous power for IFDMA with QPSK modulation and different

xiv

values of roll-off factor α. ............................................................................................................. 69 Figure 5.3: CCDF of instantaneous power for LFDMA with BPSK modulation and different values of input block size N........................................................................................................... 70 Figure 5.4: CCDF of instantaneous power for IFDMA, LFDMA, and OFDM. For IFDMA, we consider roll-off factor of 0.2. ...................................................................................................... 72 Figure 6.1: A theoretical relationship between PAPR and transmit power efficiency for ideal class A and B amplifiers. .......................................................................................................................... 75 Figure 6.2: Comparison of CCDF of PAPR for IFDMA, DFDMA, LFDMA, and OFDMA with total number of subcarriers M = 512, number of input symbols N = 128, IFDMA spreading factor Q = 4, DFDMA spreading factor Qɶ = 2, and α (roll-off factor) = 0.22.. 78 Figure 6.3: Comparison of CCDF of PAPR for IFDMA and LFDMA with M = 256, N = 64, Q = 4, Qɶ = 2, and α (roll-off factor) of 0, 0.2, 0.4, 0.6, 0.8, and 1. ............................................ 79 Figure 6.4: Precoding in the frequency domain is convolution and summation in the time domain.k refers to the subcarrier number.................................................................................... 80 Figure 6.5: CCDF of PAPR for 2x2 unitary precoded TxBF............................................................ 81 Figure 6.6: Impact of quantization and averaging of the precoding matrix on PAPR.................. 82 Figure 6.7: PAPR comparison with other MIMO schemes. .............................................................. 82 Figure 6.8: Three types of amplitude limiter........................................................................................ 85 Figure 6.9: Block diagram of a symbol amplitude clipping method for SC-FDMA MIMO transmission with Mt transmit antenna......................................................................................... 86

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Figure 6.10: CCDF of symbol power after clipping. .......................................................................... 86 Figure 6.11: Link level performance for clipping. ............................................................................... 87 Figure 6.12: PSD of the clipped signals................................................................................................ 87 Figure 7.1: Comparison of aggregate throughput with M = 256 system subcarriers, N = 8 subcarriers per user, bandwidth = 5 MHz, and noise power per Hz = -160 dBm................ 94 Figure 7.2: Average user data rate as a function of user distance with M = 256 system subcarriers, N = 8 subcarriers per user, bandwidth = 5 MHz, and noise power per Hz = -160 dBm..... 95

Figure 7.3: Block diagram of an uplink SC-FDMA system with adaptive modulation and CDS for K users. ....................................................................................................................................... 97 Figure 7.4: System throughput vs. SNR for the 8 classes of QAM................................................ 103 Figure 7.5: Aggregate throughput with CDS and adaptive modulation......................................... 104 Figure 7.6: Aggregate throughput with CDS and constant modulation (16-QAM) with mobile speed of 60 km/h.......................................................................................................................... 104 Figure 7.7: Aggregate throughput with CDS and adaptive modulation with feedback delay of 3 ms and different mobile speeds. .................................................................................................. 105 Figure 7.8: Conventional subcarrier mapping and hybrid subcarrier mapping. ............................ 107 Figure 7.9: Block diagram of an SC-CFDMA system. ..................................................................... 108 Figure 7.10: Comparison between SC-FDMA and SC-CFDMA in terms of occupied subcarriers for the same number of users...................................................................................................... 108 Figure 7.11: Aggregate throughputs for hybrid subcarrier mapping method and other conventional subcarrier mapping methods with CDS and adaptive modulation................. 109

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List of Tables Table 2.1: Transmission bandwidths of current / future cellular wireless standards..................... 15 Table 3.1: Parameters for uplink SC-FDMA transmission scheme in 3GPP LTE. ....................... 42 Table 3.2: Number of RU’s and number of subcarrriers per RU for LB........................................ 43 Table 4.1. Summary of feedback overhead vs. performance loss..................................................... 60 Table 6.1: 99.9-percentile PAPR for IFDMA, DFDMA, LFDMA, and OFDMA ........................ 78 Table 7.1: SNR boundaries for adaptive modulation........................................................................ 103

Chapter 1

Introduction

1.1.

Evolution of Cellular Wireless Communications

During the 1950s and 1960s, researchers at AT&T Bell Laboratories and companies around the world developed the idea of cellular radiotelephony. The concept of cellular wireless communications is to break the coverage zone into small cells and reuse the portions of the available radio spectrum. In 1979, the world’s first cellular system was deployed by Nippon Telephone and Telegraph (NTT) in Japan and thus began the evolution of cellular wireless communications [1], [2]. The first generation of cellular wireless communication systems utilized analog communication techniques and its focus was on accommodating voice traffic. Frequency modulation (FM) and frequency division multiple access (FDMA) were the basis of the first generation systems. AMPS (Advanced Mobile Phone System) in US and ETACS (European Total Access Cellular System) in Europe were among the first generation systems. The second generation systems saw the advent of digital communication techniques which greatly improved spectrum efficiency. Also they vastly enhanced the voice quality and made possible the packet data transmission. The main multiple access schemes are time

1

2

division multiple access (TDMA) and code division multiple access (CDMA). GSM (Global System for Mobile) which is based on TDMA and IS-95 which is based on CDMA are two most widely accepted second generation systems. In the mid-1980s, the concept for IMT-2000 (International Mobile Telecommunications2000) was born at the ITU (International Telecommunication Union) as the third generation (3G) system for mobile communications [3]. Key objectives of IMT-2000 are to provide seamless global roaming and to provide seamless delivery of services over a number of media via higher data rate link. In 2000, a unanimous approval of the technical specifications for 3G system under the brand IMT-2000 was made and UMTS/WCDMA (Universal Mobile Telecommunications System/Wideband CDMA) and cdma2000 are two prominent standards under IMT-2000 both of which are based on CDMA. IMT-2000 provides higher transmission rates; a minimum speed of 2 Mbps for stationary or walking users and 348 kbps in a moving vehicle whereas second generation systems only provide speeds ranging from 9.6 kbps to 28.8 kbps. Since the initial standardization, both WCDMA and cdma2000 have evolved into socalled “3.5G”; UMTS through HSD/UPA (High Speed Downlink/Uplink Packet Access) and cdma2000 through 1xEV-DO Rev A (1x Evolution Data-Optimized Revision A). Currently, 3rd Generation Partnership Project Long Term Evolution (3GPP LTE) is considered as the prominent path to the next generation of cellular system beyond 3G.

1.2.

3GPP Long Term Evolution

3GPP’s work on the evolution of the 3G mobile system started with the Radio Access Network (RAN) Evolution workshop in November 2004 [4]. Operators, manufacturers, and

3

research institutes presented more than 40 contributions with views and proposals on the evolution of the Universal Terrestrial Radio Access Network (UTRAN) which is the foundation for UMTS/WCDMA systems. They identified a set of high level requirements at the workshop; reduced cost per bit, increased service provisioning, flexibility of the use of existing and new frequency bands, simplified architecture and open interfaces, and allow for reasonable terminal power consumption. With the conclusions of this workshop and with broad support from 3GPP members, a feasibility study on the Universal Terrestrial Radio Access (UTRA) and UTRAN Long Term Evolution started in December 2004. The objective was to develop a framework for the evolution of the 3GPP radio access technology towards a high-data-rate, low-latency, and packet-optimized radio access technology. The study focused on means to support flexible transmission bandwidth of up to 20 MHz, introduction of new transmission schemes, advanced multi-antenna technologies, signaling optimization, identification of the optimum UTRAN network architecture, and functional split between RAN network nodes. The first part of the study resulted in an agreement on the requirements for the Evolved UTRAN (E-UTRAN). Key aspects of the requirements are as follows [5]. •

Peak data rate: Instantaneous downlink peak data rate of 100 Mbps within a 20 MHz downlink spectrum allocation (5 bps/Hz) and instantaneous uplink peak data rate of 50 Mbps (2.5 bps/Hz) within a 20 MHz uplink spectrum allocation.



Control-plane capacity: At least 200 users per cell should be supported in the active state for spectrum allocations up to 5 MHz.

4



User-plane latency: Less than 5 ms in an unloaded condition (i.e. single user with single data stream) for small IP packet.



Mobility: E-UTRAN should be optimized for low mobile speed from 0 to 15 km/h. Higher mobile speeds between 15 and 120 km/h should be supported with high performance. Mobility across the cellular network shall be maintained at speeds from 120 to 350 km/h (or even up to 500 km/h depending on the frequency band).



Coverage: Throughput, spectrum efficiency, and mobility targets should be met for 5 km cells and with a slight degradation for 30 km cells. Cells ranging up to 100 km should not be precluded.



Enhanced multimedia broadcast multicast service (E-MBMS).



Spectrum flexibility: E-UTRA shall operate in spectrum allocations of different sizes including 1.25 MHz, 1.6 MHz, 2.5 MHz, 5 MHz, 10 MHz, 15 MHz, and 20 MHz in both uplink and downlink.



Architecture and migration: Packet-based single E-UTRAN architecture with provision to support systems supporting real-time and conversational class traffic and support for an end-to-end quality-of-service (QoS).



Radio resource management: Enhanced support for end-to-end QoS, efficient support for transmission of higher layers, and support of load sharing and policy management across different radio access technologies.

The wide set of options initially identified by the early LTE work was narrowed down in

5

December 2005 to a working assumption that the downlink would use Orthogonal Frequency Division Multiple Access (OFDMA) and the uplink would use Single Carrier Frequency Division Multiple Access (SC-FDMA). Supported downlink data modulation schemes are QPSK, 16QAM, and 64QAM, and possible uplink data modulation schemes are π/2-shifted BPSK, QPSK, 8PSK and 16QAM. They agreed the use of Multiple Input Multiple Output (MIMO) scheme with possibly up to four antennas at the mobile side and four antennas at the base station. Re-using the expertise from the UTRAN, they agreed to the same channel coding type as UTRAN (turbo codes). They agreed to a transmission time interval (TTI) of 1 ms to reduce signaling overhead and to improve efficiency. The study item phase ended in September 2006 and the LTE works are scheduled to conclude in early 2008 and produce a technical standard. More technical details on 3GPP LTE are at [6] and [7].

1.3.

Single Carrier FDMA

Ever increasing demand for higher data rate is leading to utilization of wider transmission bandwidth. Broadband wireless mobile communications suffer from multipath frequencyselective fading. For broadband multipath channels, conventional time domain equalizers are impractical for complexity reason. Orthogonal frequency division multiplexing (OFDM), which is a multicarrier communication technique, has become widely accepted primarily because of its robustness against frequency-selective fading channels which are common in broadband mobile wireless communications [8]. Orthogonal frequency division multiple access (OFDMA) is a multiple

6

access scheme which is an extension of OFDM to accommodate multiple simultaneous users. OFDM/OFDMA technique is currently adopted in wireless LAN (IEEE 802.11a & 11g), WiMAX (IEEE 802.16), and 3GPP LTE downlink systems. Despite the benefits of OFDM and OFDMA, they suffer a number of drawbacks including: high peak-to-average power ratio (PAPR), a need for an adaptive or coded scheme to overcome spectral nulls in the channel, and high sensitivity to frequency offset. Single carrier frequency division multiple access (SC-FDMA) which utilizes single carrier modulation and frequency domain equalization is a technique that has similar performance and essentially the same overall complexity as those of OFDMA system [9]. One prominent advantage over OFDMA is that the SC-FDMA signal has lower PAPR because of its inherent single carrier structure. SC-FDMA has drawn great attention as an attractive alternative to OFDMA, especially in the uplink communications where lower PAPR greatly benefits the mobile terminal in terms of transmit power efficiency and manufacturing cost. SC-FDMA has two different subcarrier mapping schemes; distributed and localized. In distributed subcarrier mapping scheme, user’s data occupy a set of distributed subcarriers and we achieve frequency diversity. In localized subcarrier mapping scheme, user’s data inhabit a set of consecutive localized subcarriers and we achieve frequency-selective gain through channel-dependent scheduling (CDS). SC-FDMA is currently a working assumption for the uplink multiple access scheme in 3GPP LTE [10].

1.4. •

Objectives and Contributions Introduction to SC-FDMA: SC-FDMA is a new radio interface technique that is currently

7

being adopted in 3GPP LTE uplink. In this thesis, we give a detailed overview of an SC-FDMA system and explain its transmit and receive process. We also illustrate the similarities to and differences with an OFDMA system. •

Time domain representation of the transmit symbols of an SC-FDMA signal: In this thesis, we derive the time domain representation of the transmit symbols for each subcarrier mapping scheme.



Analysis of unitary precoded transmit eigen-beamforming (TxBF) for SC-FDMA with limited feedback: In this thesis, we numerically analyze the performance of a unitary precoded TxBF SC-FDMA system with limited feedback. We show the impacts of feedback quantization/averaging and feedback delay on the link level performance and also on the transmit PAPR characteristics.



Analytical analysis of the peak power of an SC-FDMA signal: In this thesis, we derive an analytical upper-bound on the distribution of the instantaneous power of an SCFDMA signal using Chernoff bound. We also characterize the peak power distribution for each of the subcarrier mapping scheme. We show analytically that an SC-FDMA signal has indeed lower peak power than an OFDM signal.



PAPR characteristics of an SC-FDMA signal: PAPR is an important measure that affects the transmit power efficiency. In this thesis, we numerically characterize the PAPR for different subcarrier mapping considering pulse shaping. We consider both single antenna and multiple antenna transmissions.



Peak power reduction by symbol amplitude clipping method: In this thesis, we propose a symbol

8

amplitude clipping method to reduce the peak power of the SC-FDMA transmit signal. The proposed method effectively reduces the peak power while hardly affecting the link level performance. •

Channel-dependent resource scheduling with hybrid subcarrier mapping: Channel-dependent scheduling (CDS) results in high capacity gain when channel state information (CSI) is accurate but the gain decreases when the quality of CSI becomes poor. In this thesis, we propose a hybrid subcarrier mapping scheme utilizing orthogonal code spreading for cases where there are both high and low mobility users at the same time. Our proposed scheme yields higher capacity gain when high mobility users are dominant.

1.5.

Organization

The following is the outline of the remainder of the thesis. Chapter 2 gives a general overview of OFDM and frequency domain equalization. We first characterize the wireless mobile communications channel. Then, we give an overview of OFDM which is a popular multicarrier modulation technique, and we explain single carrier modulation with frequency domain equalization (SC/FDE) and compare it with OFDM. Chapter 3 introduces SC-FDMA. We first give an overview of SC-FDMA and explain the transmission and reception operations in detail. We describe the two flavors of subcarrier mapping schemes in SC-FDMA, distributed and localized, and briefly compare the two, and we derive the time domain representation of SC-FDMA transmit signal for each subcarrier mapping mode. Then, we give an in-depth comparison between SC-FDMA and OFDMA and we also compare SC-FDMA with direct sequence spread spectrum code division multiple

9

access (DS-CDMA) with frequency domain equalization. Lastly, we illustrate in detail the SCFDMA implementation in the physical layer according to 3GPP LTE uplink and we describe the reference signal structure of SC-FDMA. Chapter 4 investigates MIMO techniques for an SC-FDMA system. We first give a general overview of MIMO concepts and describe the parallel decomposition of a MIMO channel for narrowband and wideband transmission. Then, we illustrate the realization of MIMO spatial multiplexing in SC-FDMA. We introduce the SC-FDMA TxBF with unitary precoding technique and numerically analyze the link level performance with practical considerations. Chapter 5 analyzes the distribution of the instantaneous peak power of an SC-FDMA signal and shows analytically that an SC-FDMA signal has statistically lower peak power than an OFDM signal. Using the Chernoff bound, we first derive an upper-bound of the complementary cumulative distribution function (CCDF) of the instantaneous power for IFDMA with pulse shaping and show the bounds for BPSK and QPSK with raised-cosine pulse shaping filter. We also derive a modified upper-bound of the CCDF of the instantaneous power for LFDMA without pulse shaping. Then, we compare the results with the analytical CCDF of PAPR for OFDM. Chapter 6 investigates the PAPR characteristics of an SC-FDMA signal numerically. We first characterize the PAPR for single antenna transmission of SC-FDMA. We investigate the PAPR properties for different subcarrier mapping schemes. Then, we analyze the PAPR characteristics for multiple antenna transmission. Specifically, we numerically analyze the CCDF of PAPR for a 2x2 unitary precoded TxBF SC-FDMA system described in chapter 4.

10

Lastly, we propose a symbol amplitude clipping method to reduce peak power and show the link level performance and frequency domain aspects of the proposed clipping method. Chapter 7 presents a channel-dependent scheduling (CDS) method for an SC-FDMA system in the uplink communications. We first give a general overview of CDS in an uplink SC-FDMA system. We also analyze the capacity for the two subcarrier mapping schemes. Then, we investigate the impact of imperfect channel state information (CSI) on CDS. We analyze the data throughput of an SC-FDMA system with uncoded adaptive modulation and CDS when there is a feedback delay. Lastly, we propose a hybrid subcarrier mapping scheme using direct sequence spreading technique on top of SC-FDMA modulation. We show the throughput improvement of the hybrid subcarrier mapping over the conventional subcarrier mapping schemes. Chapter 8 presents a summary of the work and concluding remarks.

1.6.

Nomenclature 3GPP

3rd Generation Partnership Project

BER

Bit Error Rate

BPSK

Binary Phase Shift Keying

CAZAC

Constant Amplitude Zero Auto-Correlation

CCDF

Complementary Cumulative Distribution Function

CDM

Code Division Multiplexing

CDMA

Code Division Multiple Access

CDS

Channel-Dependent Scheduling

11

CP

Cyclic Prefix

CSI

Channel State Information

DFDMA

Distributed Frequency Division Multiple Access

DFT

Discrete Fourier Transform

FDE

Frequency Domain Equalization

FDM

Frequency Division Multiplexing

FDMA

Frequency Division Multiple Access

FER

Frame Error Rate

FFT

Fast Fourier Transform

GSM

Global System for Mobile

IBI

Inter-Block Interference

ICI

Inter-Carrier Interference

IDFT

Inverse Discrete Fourier Transform

ISI

Inter-Symbol Interference

IFDMA

Interleaved Frequency Division Multiple Access

LB

Long Block

LFDMA

Localized Frequency Division Multiple Access

LMMSE

Linear Minimum Mean Square Error

LTE

Long Term Evolution

MIMO

Multiple Input Multiple Output

MMSE

Minimum Mean Square Error

12

OFDM

Orthogonal Frequency Division Multiplexing

OFDMA

Orthogonal Frequency Division Multiple Access

PAPR

Peak-to-Average Power Ratio

PSD

Power Spectral Density

QAM

Quadrature Amplitude Modulation

QPSK

Quadrature Phase Shift Keying

RU

Resource Unit

SB

Short Block

SC

Single Carrier

SC/FDE

Single Carrier with Frequency Domain Equalization

SC-FDMA

Single Carrier Frequency Division Multiple Access

SFBC

Space-Frequency Block Coding

SM

Spatial Multiplexing

SNR

Signal-to-Noise Ratio

SVD

Singular Value Decomposition

TDM

Time Division Multiplexing

TDMA

Time Division Multiple Access

TTI

Transmission Time Interval

TxBF

Transmit Eigen-Beamforming

WCDMA

Wideband Code Division Multiple Access

Chapter 2

Channel Characteristics and Frequency Multiplexing Multiplexing

In this chapter, we first characterize the wireless mobile communications channel. In section 2.2, we give an overview of orthogonal frequency division multiplexing (OFDM) which is a popular multicarrier modulation technique and in section 2.3, we explain single carrier modulation with frequency domain equalization (SC/FDE) and compare it with OFDM.

2.1.

Characteristics of Wireless Mobile Communications Channel

In a wireless mobile communication system, a transmitted signal propagating through the wireless channel often encounters multiple reflective paths until it reaches the receiver [11]. We refer to this phenomenon as multipath propagation and it causes fluctuation of the amplitude and phase of the received signal. We call this fluctuation multipath fading and it can occur either in large scale or in small scale. Large-scale fading represents the average signal power attenuation or path loss due to motion over large areas. Small-scale fading occurs due to small changes in position and we also call it as Rayleigh fading since the fading is often statistically characterized with Rayleigh probability density function (pdf). Rayleigh fading in the propagation channel, which generates inter-symbol interference (ISI) in the time domain, is a major impairment in wireless communications and it significantly degrades the link performance.

13

14 3GPP 6-Tap Typical Urban (TU6) Channel Delay Profile

Frequency Response of 3GPP TU6 Channel in 5MHz Band 2.5

1 2

Channel Gain [linear]

Amplitude [linear]

0.8

0.6

0.4

1

0.5

0.2

0

1.5

0

1

2

3 Time [ µsec]

4

5

6

0

0

1

2 3 Frequency [MHz]

4

5

Figure 2.1: Delay profile and frequency response of 3GPP 6-tap typical urban (TU6) [12] Rayleigh fading channel in 5 MHz band. When characterizing the Rayleigh fading channel, we can categorize it into either flat fading channel or frequency-selective fading channel. Flat fading occurs when the coherence bandwidth, which is inversely proportional to channel delay spread, is much larger than the transmission bandwidth whereas frequency-selective fading happens when the coherence bandwidth is much smaller than the transmission bandwidth. We show an example of the impulse response and frequency response of a frequency-selective fading channel in Figure 2.1. We can also characterize the multipath fading channel in terms of the degree of time variation of the channel; slow fading and fast fading. The time-varying nature of the channel is directly related to the movement of the user and the user’s surrounding, and the degree of the time variation is associated with the Doppler frequency. Doppler frequency fd is given by fd =

v

λ

(2.1)

15

where v is the relative speed of the user and λ is the wavelength of the carrier. For a given Doppler frequency fd, the spaced-time correlation function R(∆t) specifies the extent to which there is correlation between the channel’s response in ∆t time interval and it is given by

R ( ∆t ) = J 0 ( 2π f d ∆t )

(2.2)

where J0(⋅) is the zero-order Bessel function of the first kind. As the Doppler frequency increases, the correlation decreases at a given time interval. An example of the time variation of a fading channel is illustrated in Figure 2.2. As wireless multimedia applications become more wide-spread, demand for higher data rate is leading to utilization of a wider transmission bandwidth. For example, Global System for Mobile Communications (GSM) system, which is a popular second generation cellular system, uses transmission bandwidth of only 200 kHz but the next generation cellular standard 3GPP Long Term Evolution (LTE) envisions bandwidth of up to 20 MHz which is 100 times the bandwidth of GSM. Table 2.1 illustrates the transmission bandwidth in current and future cellular wireless standards. Table 2.1: Transmission bandwidths of current / future cellular wireless standards. Generation 2G 3G 3.5 ~ 4G

Standard

Transmission Bandwidth

GSM

200 kHz

IS-95 (CDMA)

1.25 MHz

WCDMA

5 MHz

cdma2000

5 MHz

3GPP LTE

Up to 20 MHz

WiMAX (IEEE 802.16)

Up to 20 MHz

16

Channel Gain [linear]

Mobile speed = 3 km/h (5.6 Hz doppler)

5

0 5

5 4

4 3

3 2

2 1

Frequency [MHz]

1 0

Time [msec]

0

(a)

Channel Gain [linear]

Mobile speed = 60 km/h (111 Hz doppler)

5

0 5

5 4

4 3

3 2

2 1

Frequency [MHz]

1 0

0

Time [msec]

(b) Figure 2.2: Time variation of 3GPP TU6 Rayleigh fading channel in 5 MHz band with 2GHz carrier frequency; (a) user speed = 3 km/h (Doppler frequency = 5.6 Hz); (b) user speed = 60 km/h (Doppler frequency = 111 Hz).

17

With a wider transmission bandwidth, frequency selectivity of the channel becomes more severe and thus the problem of ISI becomes more serious. In a conventional single carrier communication system, time domain equalization in the form of tap delay line filtering is performed to eliminate ISI. However, in case of a wide band channel, the length of the time domain filter to perform equalization becomes prohibitively large since it linearly increases with the channel response length.

2.2.

Orthogonal Frequency Division Multiplexing (OFDM)

One way to mitigate the frequency-selective fading seen in a wide band channel is to use a multicarrier technique which subdivides the entire channel into smaller sub-bands, or subcarriers. Orthogonal frequency division multiplexing (OFDM) is a multicarrier modulation technique which uses orthogonal subcarriers to convey information. In the frequency domain, since the bandwidth of a subcarrier is designed to be smaller than the coherence bandwidth, each subchannel is seen as a flat fading channel which simplifies the channel equalization process. In the time domain, by splitting a high-rate data stream into a number of lower-rate data stream that are transmitted in parallel, OFDM resolves the problem of ISI in wide band communications. More technical details on OFDM are at [8], [13], [14], [15], [16], and [17]. In summary, OFDM has the following advantages: •

For a given channel delay spread, the implementation complexity is much lower than that of a conventional single carrier system with time domain equalizer.



Spectral efficiency is high since it uses overlapping orthogonal subcarriers in the frequency domain.

18



Modulation and demodulation are implemented using inverse discrete Fourier transform (IDFT) and discrete Fourier transform (DFT), respectively, and fast Fourier transform (FFT) algorithms can be applied to make the overall system efficient.



Capacity can be significantly increased by adapting the data rate per subcarrier according to the signal-to-noise ratio (SNR) of the individual subcarrier.

Because of these advantages, OFDM has been adopted as a modulation of choice by many wireless communication systems such as wireless LAN (IEEE 802.11a and 11g) and DVB-T (Digital Video Broadcasting-Terrestrial). However, it suffers from the following drawbacks [18], [19]: •

High peak-to-average power ratio (PAPR): The transmitted signal is a superposition of all the subcarriers with different carrier frequencies and high amplitude peaks occur because of the superposition.



High sensitivity to frequency offset: When there are frequency offsets in the subcarriers, the orthogonality among the subcarriers breaks and it causes intercarrier interference (ICI).



A need for an adaptive or coded scheme to overcome spectral nulls in the channel: In the presence of a null in the channel, there is no way to recover the data of the subcarriers that are affected by the null unless we use rate adaptation or a coding scheme.

19 SC/FDE

{ xn }

Add CP/ PS

Channel

Remove CP

Npoint DFT

Equalization

Add CP/ PS

Channel

Remove CP

Npoint DFT

Equalization

Npoint IDFT

Detect

OFDM

Npoint IDFT

{ xn }

Detect

* CP: Cyclic Prefix, PS: Pulse Shaping

Figure 2.3: Transmitter and receiver structures of SC/FDE and OFDM.

2.3.

Single Carrier with Frequency Domain Equalization (SC/FDE)

For broadband multipath channels, conventional time domain equalizers are impractical because of the complexity (very long channel impulse response in the time domain). Frequency domain equalization (FDE) is more practical for such channels. Single carrier with frequency domain equalization (SC/FDE) technique is another way to fight the frequency-selective fading channel. It delivers performance similar to OFDM with essentially the same overall complexity, even for long channel delay [18], [19]. Figure 2.3 shows the block diagram of SC/FDE and compares it with that of OFDM. In the transmitter of SC/FDE, we add a cyclic prefix (CP), which is a copy of the last part of the block, to the input data at the beginning of each block in order to prevent inter-block

20

interference (IBI) and also to make linear convolution of the channel impulse response look like a circular convolution. It should be noted that circular convolution problem exists for any FDE since multiplication in the DFT-domain is equivalent to circular convolution in the time domain [20]. When the data signal propagates through the channel, it linearly convolves with the channel impulse response. An equalizer basically attempts to invert the channel impulse response and thus channel filtering and equalization should have the same type of convolution, either linear or circular convolution. One way to resolve this problem is to add a CP in the transmitter that will make the channel filtering look like a circular convolution and match the DFT-based FDE. Another way is not to use CP but perform an “overlap and save” method in the frequency domain equalizer to emulate the linear convolution [20]. SC/FDE receiver transforms the received signal to the frequency domain by applying DFT and does the equalization process in the frequency domain. Most of the well-known time domain equalization techniques, such as minimum mean-square error (MMSE) equalization, decision feedback equalization, and turbo equalization, can be applied to the FDE and the details of the frequency domain implementation of these techniques are found in [21], [22], [23], [24], [25], and [26]. After the equalization, the signal is brought back to the time domain via IDFT and detection is performed. Comparing the two systems in Figure 2.3, it is interesting to find the similarity between the two. Overall, they both use the same communication component blocks and the only difference between the two diagrams is the location of the IDFT block. Thus, one can expect the two systems to have similar link level performance and spectral efficiency.

21

OFDM

SC/FDE

DFT

DFT

Equalizer

Detect

Equalizer

Detect

Equalizer

Detect

Equalizer

IDFT

Detect

(a)

OFDM symbol SC/FDE symbols time

(b) Figure 2.4: Dissimilarities between OFDM and SC/FDE; (a) different detection processes in the receiver; (b) different modulated symbol durations. However, there are distinct differences that make the two systems perform differently as illustrated in Figure 2.4. In the receiver, OFDM performs data detection on a per-subcarrier basis in the frequency domain whereas SC/FDE does it in the time domain after the additional IDFT operation. Because of this difference, OFDM is more sensitive to a null in the channel spectrum and it requires channel coding or power/rate control to overcome this deficiency. Also, the duration of the modulated time symbols are expanded in the case of OFDM with parallel transmission of the data block during the elongated time period.

22

In summary, SC/FDE has advantages over OFDM as follows: •

Low PAPR due to single carrier modulation at the transmitter.



Robustness to spectral null.



Lower sensitivity to carrier frequency offset.



Lower complexity at the transmitter which will benefit the mobile terminal in cellular uplink communications.

Single carrier FDMA (SC-FDMA) is an extension of SC/FDE to accommodate multi-user access, which will be the subject of the next chapter.

2.4.

Summary and Conclusions

Broadband mobile wireless channel suffers from severe frequency-selective fading which causes the variation of received signal strength. OFDM is a multicarrier technique that overcomes the frequency-selective fading impairment by transmitting data over narrower subbands in parallel. Despite the many benefits, OFDM has limits including: high peak-to-average power ratio (PAPR), high sensitivity to frequency offset, and a need for an adaptive or coded scheme to overcome spectral nulls in the channel. Single carrier modulation with frequency domain equalization (SC/FDE) technique is another way to mitigate the frequency-selective fading. SC/FDE delivers performance similar to OFDM with essentially the same overall complexity and has advantages including; low PAPR, robustness to spectral null, lower sensitivity to carrier frequency offset, lower complexity at the transmitter which will benefit the mobile terminal in cellular uplink communications.

Chapter 3

Single Carrier FDMA

Single carrier frequency division multiple access (SC-FDMA), which utilizes single carrier modulation and frequency domain equalization, is a technique that has similar performance and essentially the same overall complexity as those of orthogonal frequency division multiple access (OFDMA) system. SC-FDMA is an extension of single carrier modulation with frequency domain equalization (SC/FDE) to accommodate multiple-user access. One prominent advantage over OFDMA is that the SC-FDMA signal has lower peak-to-average power ratio (PAPR) because of its inherent single carrier structure [9]. SC-FDMA has drawn great attention as an attractive alternative to OFDMA, especially in the uplink communications where lower PAPR greatly benefits the mobile terminal in terms of power efficiency. It is currently a working assumption for uplink multiple access scheme in 3GPP Long Term Evolution (LTE) or Evolved UTRA [6], [7], [10]. In this chapter, we first give an overview of SC-FDMA and explain the transmission and reception operations in detail. In section 3.2, we describe the two flavors of subcarrier mapping schemes in SC-FDMA and briefly compare the two. In section 3.3, we derive the time domain representations of SC-FDMA transmit signal for each subcarrier mapping mode.

23

24 SC-FDMA Npoint DFT

{ xn }

{X k }

Subcarrier Mapping

M-

point { Xɶ } IDFT l

{ xɶm }

Add CP / PS

DAC / RF

Channel

Detect

Npoint IDFT

Subcarrier Demapping/ Equalization

Mpoint DFT

Remove CP

RF / ADC

Subcarrier Mapping

Mpoint IDFT

Add CP / PS

DAC / RF

OFDMA

{ xn }

Channel

Detect

Subcarrier Demapping/ Equalization

Mpoint DFT

Remove CP

RF / ADC

* CP: Cyclic Prefix, PS: Pulse Shaping

Figure 3.1: Transmitter and receiver structure of SC-FDMA and OFDMA systems. In section 3.4, we give an in-depth comparison between SC-FDMA and OFDMA. In section 3.5, we compare SC-FDMA with direct sequence spread spectrum code division multiple access (DS-CDMA) with frequency domain equalization and show the similarities between the two. In section 3.6, we illustrate in detail the SC-FDMA implementation in the physical layer according to 3GPP LTE uplink. In section 3.7, we describe the reference (pilot) signal structure of SC-FDMA.

25

3.1.

Overview of SC-FDMA System

Figure 3.1 shows a block diagram of an SC-FDMA system. SC-FDMA can be regarded as DFT-spread OFDMA, where time domain data symbols are transformed to frequency domain by DFT before going through OFDMA modulation. The orthogonality of the users stems from the fact that each user occupies different subcarriers in the frequency domain, similar to the case of OFDMA. Because the overall transmit signal is a single carrier signal, PAPR is inherently low compared to the case of OFDMA which produces a multicarrier signal. The transmitter of an SC-FDMA system converts a binary input signal to a sequence of modulated subcarriers. At the input to the transmitter, a baseband modulator transforms the binary input to a multilevel sequence of complex numbers xn in one of several possible modulation formats. The transmitter next groups the modulation symbols {xn} into blocks each containing N symbols. The first step in modulating the SC-FDMA subcarriers is to perform an N-point DFT to produce a frequency domain representation Xk of the input symbols. It then maps each of the N DFT outputs to one of the M (> N) orthogonal subcarriers that can be transmitted. If N = M/Q and all terminals transmit N symbols per block, the system can handle Q simultaneous transmissions without co-channel interference. Q is the bandwidth expansion factor of the symbol sequence. The result of the subcarrier mapping is the set Xɶ l (l = 0, 1, 2…, M-1) of complex subcarrier amplitudes, where N of the amplitudes are non-zero. As in OFDMA, an M-point IDFT transforms the subcarrier amplitudes to a complex time domain signal xɶm . Each xɶm then are transmitted sequentially. The transmitter performs two other signal processing operations prior to transmission. It

26

inserts a set of symbols referred to as a cyclic prefix (CP) in order to provide a guard time to prevent inter-block interference (IBI) due to multipath propagation. The transmitter also performs a linear filtering operation referred to as pulse shaping in order to reduce out-ofband signal energy. In general, CP is a copy of the last part of the block, which is added at the start of each block for a couple of reasons. First, CP acts as a guard time between successive blocks. If the length of the CP is longer than the maximum delay spread of the channel, or roughly, the length of the channel impulse response, then, there is no IBI. Second, since CP is a copy of the last part of the block, it converts a discrete time linear convolution into a discrete time circular convolution. Thus transmitted data propagating through the channel can be modeled as a circular convolution between the channel impulse response and the transmitted data block, which in the frequency domain is a point-wise multiplication of the DFT frequency samples. Then, to remove the channel distortion, the DFT of the received signal can simply be divided by the DFT of the channel impulse response point-wise or a more sophisticated frequency domain equalization technique can be implemented. One of the commonly used pulse shaping filter is the raised-cosine filter. The frequency domain and time domain representations of the filter are as follows.

1−α  ,0 ≤ f ≤ T 2T  πT  1 − α   1 − α 1+ α T  P( f ) =  1 + cos  ≤ f ≤  f −   , 2T    2T 2T α  2   1+α 0 , f ≥  2T

(3.1)

27 P(f)

p(t)

1

0.8

α=0 α = 0.5

0.8

α=1

0.6

0.6 0.4 0.4

α=1

0.2 0

0.2

-0.2 0

Frequency

α=0 Time

α = 0.5

Figure 3.2: Raised-cosine filter.

p(t ) =

sin (π t / T ) cos (πα t / T ) ⋅ πt /T 1 − 4α 2t 2 / T 2

(3.2)

where T is the symbol period and α is the roll-off factor. Figure 3.2 shows the raised-cosine filter graphically in the frequency domain and time domain. Roll-off factor α changes from 0 to 1 and it controls the amount of out-of-band radiation; α = 0 generates no out-of-band radiation and as α increases, the out-of-band radiation increases. In the time domain, the pulse has higher side lobes when α is close to 0 and this increases the peak power for the transmitted signal after pulse shaping. We further investigate the effect of pulse shaping on the peak power characteristics in chapters 5 and 6. Figure 3.3 details the generation of SC-FDMA transmit symbols. There are M subcarriers, among which N (< M) subcarriers are occupied by the input data. In the time domain, the input data symbol has symbol duration of T seconds and the symbol duration is compressed

28

{ xn }

DFT (N-point)

N Tɶ

{X k }

Subcarrier Mapping

N Tɶ

{ Xɶ } l

IDFT (M-point)

M >N N T = Tɶ ⋅ M

{ xɶm } M T

N , M : number of data symbols Tɶ , T : symbol durations

Figure 3.3: Generation of SC-FDMA transmit symbols. There are M total number of subcarriers, among which N (< M) subcarriers are occupied by the input data. to Tɶ = (N/M)⋅T seconds after going through SC-FDMA modulation. The receiver transforms the received signal into the frequency domain via DFT, de-maps the subcarriers, and then performs frequency domain equalization. Because SC-FDMA uses single carrier modulation, it suffers from inter-symbol interference (ISI) and thus equalization is necessary to combat the ISI. The equalized symbols are transformed back to the time domain via IDFT, and detection and decoding take place in the time domain.

3.2.

Subcarrier Mapping

There are two methods to choose the subcarriers for transmission as shown in Figure 3.4; distributed subcarrier mapping and localized subcarrier mapping. In the distributed subcarrier mapping mode, DFT outputs of the input data are allocated over the entire bandwidth with zeros occupying the unused subcarriers, whereas consecutive subcarriers are occupied by the DFT outputs of the input data in the localized subcarrier mapping mode. We will refer to the localized subcarrier mapping mode of SC-FDMA as localized FDMA (LFDMA) and

29

Xɶ 0

X0 Zeros

Xɶ 0

Zeros

X0 X1

X1 Zeros

X2 X N −1 X N −1

Zeros Zeros

Xɶ M −1

Xɶ M −1

Distributed Mode

Localized Mode

Figure 3.4: Subcarrier mapping modes; distributed and localized. distributed subcarrier mapping mode of SC-FDMA as distributed FDMA (DFDMA). The case of M = Q⋅N for the distributed mode with equidistance between occupied subcarriers is called Interleaved FDMA (IFDMA) [27], [28], [24]. IFDMA is a special case of SC-FMDA and it is very efficient in that the transmitter can modulate the signal strictly in the time domain without the use of DFT and IDFT. An example of SC-FDMA transmit symbols in the frequency domain for N = 4, Q = 3 and M = 12 is illustrated in Figure 3.5 and a multi-user perspective is shown in Figure 3.6. From a resource allocation point of view, subcarrier mapping methods are further divided into static and channel-dependent scheduling (CDS) methods. CDS assigns subcarriers to users according to the channel frequency response of each user. For both scheduling methods, distributed subcarrier mapping provides frequency diversity because the transmitted signal is

30

M = Q—N 12= 3—4

{X~

M > Q—N 12> 2—4 M = Q—N 12= 3—4

{ xn } :

x0

x1

x2

{X k } :

X0

X1

X2

X3

X0

0

0

X1

0

0

X2

0

0

X3

0

0

X0

0

X1

0

X2

0

X3

0

0

0

0

0

X0

X1

X2

X3

0

0

0

0

0

0

0

0

l , IFDMA

{X~ {X~

}

l , DFDMA

l , LFDMA

} }

x3

2π N −1 −j nk  DFT  X k = ∑ xn e N n=0 

 , N = 4 

frequency *M : Total number of subcarriers, N : Data block size, Q : Bandwidth spreading factor

Figure 3.5: An example of different subcarrier mapping schemes for N = 4, Q = 3 and M = 12.

Terminal 1 Terminal 2 Terminal 3

subcarriers Distributed Mode

subcarriers Localized Mode

Figure 3.6: Subcarrier allocation methods for multiple users (3 users, 12 subcarriers, and 4 subcarriers allocated per user).

31

spread over the entire bandwidth. With distributed mapping, CDS incrementally improves performance. By contrast, CDS is of great benefit with localized subcarrier mapping because it provides significant multi-user diversity. We will discuss this aspect in more detail in chapter 7.

3.3.

Time Domain Representation of SC-FDMA Signals

We derive the time domain symbols without pulse shaping for each subcarrier mapping scheme. In the subsequent derivations, we will follow the notations in Figure 3.3. 3.3.1. Time Domain Symbols of IFDMA For IFDMA, the frequency samples after subcarrier mapping  X l / Q , l = Q ⋅ k Xɶ l =   0 , otherwise

(0 ≤ k ≤ N − 1)

{ Xɶ l } can be described as follows. (3.3)

where 0 ≤ l ≤ M − 1 and M = Q ⋅ N . Let m = N ⋅ q + n ( 0 ≤ q ≤ Q − 1 , 0 ≤ n ≤ N − 1 ). Then,

1 xɶm ( = xɶ Nq + n ) = M =

M −1

m j 2π l 1 1 M ɶ Xle = ⋅ ∑ Q N l =0

1 1 ⋅ Q N

N −1

∑ Xke N −1

∑X e k =0

j 2π

k

∑X e k =0

j 2π

m k N

k

Nq + n k N

k =0

1 1 = ⋅ Q N =

j 2π

N −1

n k N

  

(3.4)

1 1 xn = x( m ) mod N Q Q

The resulting time symbols { xɶm } are simply a repetition of the original input symbols

32

{ xn } with a scaling factor of 1/Q in the time domain. When the subcarrier allocation starts from rth subcarrier ( 0 < r ≤ Q − 1 ), then,  Xɶ l =  

X l / Q−r

, l = Q ⋅ k + r (0 ≤ k ≤ N − 1)

(3.5)

0 , otherwise

xɶm ( = xɶ Nq + n ) = =

1 M

M −1

m j 2π l 1 1 M ɶ X e = ⋅ ∑ l Q N l =0

1 1 ⋅ Q N

N −1

∑ X ke

e

∑ X ke

j 2π

 m mr  j 2π  k +  M  N

k =0 mr M

k =0

1 1 = ⋅ Q N =

Nq + n j 2π k N

N −1

N −1

∑X k =0

k

e

j 2π

n k N

(3.6)

 j 2π mr ⋅e M 

1 j 2π mr 1 j 2π mr e M ⋅ xn = e M ⋅ x( m ) mod N Q Q

Thus, there is an additional phase rotation of e

j 2π

mr M

when the subcarrier allocation

starts from rth subcarrier instead of subcarrier zero. This phase rotation will also apply to the other subcarrier mapping schemes as well in the same case. 3.3.2. Time Domain Symbols of LFDMA For LFDMA, the frequency samples after subcarrier mapping { Xɶ l } can be described as follows.  X , 0 ≤ l ≤ N −1 Xɶ l =  l 0 , N ≤ l ≤ M −1

(3.7)

Let m = Q ⋅ n + q , where 0 ≤ n ≤ N − 1 and 0 ≤ q ≤ Q − 1 . Then,

xɶm = xɶQn + q

1 = M

M −1

m j 2π l 1 1 M ɶ X e = ⋅ ∑ l Q N l =0

N −1

∑X e l =0

l

j 2π

Qn + q l QN

(3.8)

33

If q = 0, then, Qn

j 2π l 1 1 N −1 1 1 ⋅ ∑ X l e QN = ⋅ Q N l =0 Q N 1 1 = xn = x( m ) mod N Q Q

xɶm = xɶQn =

N −1

If q ≠ 0, since X l = ∑ x p e

− j 2π

p l N

N −1

∑ X le

j 2π

n l N

l =0

(3.9)

, then (3.8) can be expressed as follows.

p=0

xɶm = xɶQ⋅n+ q 1 1 = ⋅ Q N =

=

1 1 ⋅ Q N

1 1 = ⋅ Q N

N −1

∑X e

∑∑ x e ∑x p =0

1  ⋅ 1 − e Q 

Q⋅ N

l

1 1 = ⋅ Q N

 ( n− p ) q  + j 2π  l Q⋅ N   N

p

l =0 p = 0

1− e

N −1

Q⋅n +q

l

l =0

N −1 N −1

j 2π

p

j 2π ( n − p )

e

q Q

 (n− p) q  + j 2π   Q⋅ N   N

1− e  1 ⋅   N

q j 2π Q

j 2π

N −1

1 1 = ⋅ Q N =

1 1 ⋅ Q N

 N −1 j 2π  ( n − p ) + q l  x p  ∑ e  N Q⋅ N   ∑  l =0  p =0   N −1

1− e

N −1

∑x p =0

p

1− e

j 2π

q Q

(3.10)

 (n− p) q  + j 2π   Q⋅ N   N

xp

∑ p =0

 N −1 − j 2π p l  j 2π QQ⋅n⋅+Nq l N  ∑ x p e  e ∑ l =0  p =0  N −1

1− e

 (n− p) q  j 2π  +  Q⋅ N   N

As can be seen from (3.9) and (3.10), LFDMA signal in the time domain has exact copies of input time symbols with a scaling factor of 1/Q in the N-multiple sample positions and inbetween values are sum of all the time input symbols in the input block with different complex-weighting. 3.3.3. Time Domain Symbols of DFDMA For DFDMA, the frequency samples after subcarrier mapping { Xɶ l } can be described as follows.

34  X ɶ , l = Qɶ ⋅ k Xɶ l =  l / Q 0 , otherwise

(0 ≤ k ≤ N − 1)

(3.11)

where 0 ≤ l ≤ M − 1 , M = Q ⋅ N , and 1 ≤ Qɶ < Q . Let m = Q ⋅ n + q ( 0 ≤ n ≤ N − 1 , 0 ≤ q ≤ Q − 1 ). Then,

xɶm (= xɶQ⋅n + q ) =

1 M −1 ɶ j 2π Mm l ⋅ ∑ Xle M l =0

j 2π 1 1 N −1 = ⋅ ∑ X ke Q N k =0

(3.12)

Qn + q ɶ Qk QN

If q = 0, then, Qn

ɶ ɶ j 2π Qk j 2π Qk 1 1 N −1 1 1 N −1 xɶm = xɶQ⋅n = ⋅ ∑ X k e QN = ⋅ ∑ X ke N Q N k =0 Q N k =0 (Qɶ ⋅n )mod N  Qɶ ⋅n j 2π k j 2π k 1 1 N −1 1  1 N −1 N N  = ⋅ ∑ X ke = ⋅∑ Xke Q N k =0 Q  N k =0    1 1 = ⋅ x(Qɶ ⋅n ) = ⋅ x( Qɶ ( m ) ) mod N mod N mod N Q Q N −1

If q ≠ 0, since X k = ∑ x p e

− j 2π

p k N

n

(3.13)

, (3.12) can be expressed as follows after derivation.

p=0

xɶm = xɶQ⋅n + q

Qɶ j 2π q  1 1 Q ⋅ = 1 − e  Q   N

N −1

x (

∑ p=0

1− e

p ɶ −p  Qn ɶ  Qq j 2π  +  QN   N 

)

(3.14)

It is interesting to see that the time domain symbols of DFDMA have the same structure as those of LFDMA. Figure 3.7 shows an example of the time symbols for each subcarrier mapping mode

35

{ xn }

x0

x1

x2

x3

{Q ⋅ xɶ

}

x 0 x1

x2 x 3 x 0 x1 x2 x 3 x 0 x1 x2 x 3

{Q ⋅ xɶ

m , LFDMA

}

x0 *

* x1 * * x2 * * x 3 * *

{Q ⋅ xɶ

m , DFDMA

}

x0 *

* x2 * * x0 * * x 2 * *

m , IFDMA

time 3

* = ∑ ck ,m ⋅ xk k =0

, ck ,m : complex weight

Figure 3.7: Time symbols of different subcarrier mapping schemes. 0.5 IFDMA LFDMA DFDMA

Amplitude [linear]

0.4

0.3

0.2

0.1

0

10

20

30 Symbol

Figure 3.8: Amplitude of SC-FDMA signals.

40

50

60

36

based on Figure 3.5 for M = 12, N = 4, Q = 3, and Qɶ = 2. Figure 3.8 shows the amplitude of the signal for each subcarrier mapping for M = 64, N = 16, Q = 4, and Qɶ = 3 and we can see more fluctuation and higher peak for LFDMA and DFDMA.

3.4.

SC-FDMA and OFDMA

Figure 3.1 includes a block diagram of an OFDMA transmitter. It has much in common with SC-FDMA. The only difference is the presence of the DFT in SC-FDMA. For this reason SCFDMA is sometimes referred to as DFT-spread or DFT-precoded OFDMA. Other similarities between the two include: block-based data modulation and processing, division of the transmission bandwidth into narrower sub-bands, frequency domain channel equalization process, and the use of CP. However, there are distinct differences that make the two systems perform differently as illustrated in Figure 3.9. In terms of data detection at the receiver, OFDMA performs it on a per-subcarrier basis whereas SC-FDMA does it after additional IDFT operation. Because of this difference, OFDMA is more sensitive to a null in the channel spectrum and it requires channel coding or power/rate control to overcome this deficiency. Also, the duration of the modulated time symbols are expanded in the case of OFDMA with parallel transmission of the data block during the elongated time period whereas SC-FDMA modulated symbols are compressed into smaller chips with serial transmission of the data block, much like a direct sequence code division multiple access (DS-CDMA) system.

37

OFDMA

SC-FDMA

DFT

DFT

Subcarrier Demapping

Subcarrier Demapping

Equalizer

Detect

Equalizer

Detect

Equalizer

Detect

Equalizer

IDFT

Detect

(a)

Input data symbols OFDMA symbol SC-FDMA symbols* * Bandwidth spreading factor : 4

time

(b) Figure 3.9: Dissimilarities between OFDMA and SC-FDMA; (a) different detection processes in the receiver; (b) different modulated symbol durations.

38

3.5.

SC-FDMA and DS-CDMA/FDE

Direct sequence code division multiple access (DS-CDMA) with FDE is a technique that replaces the rake combiner, commonly used in the conventional DS-CDMA, with the frequency domain equalizer [29]. A rake receiver consists of a bank of correlators, each of which correlate to a particular multipath component of the desired signal. As the number of multipaths increase, the frequency selectivity in the channel also increases and the complexity of the rake combiner increases since more correlators are needed. The use of FDE instead of rake combing can alleviate the complexity problem in DS-CDMA. Block diagram of DSCDMA/FDE is shown in Figure 3.10.

{ xn }

Spreading

Add CP/ PS

Channel

Remove CP

Mpoint DFT

Equalization

Mpoint IDFT

Despreading

Detect

Figure 3.10: DS-CDMA with FDE. The transmitter of DS-CDMA/FDE is the same as the conventional DS-CDMA except for the addition of CP. The FDE in the receiver removes the channel distortion from the received chip symbols to recover ISI-free chip symbols. For small spreading factors, the link performance of the rake receiver significantly degrades because of the inter-path interference and FDE has a much better performance. For large spreading factors, both have similar performances. More technical details of the DS-CDMA/FDE system can be found in [29]. SC-FDMA is similar to DS-CDMA/FDE in terms of the following aspects:

39

x0

x1

x2

×

Data Sequence

Conventional Spreading

1

1

1

x3

1

1

1

1

1

1

1

1

1

1

1

1

1

Signature Sequence

x0 x0

x0 x0 x1 x1

x 1 x 1 x 2 x 2 x2 x2 x 3 x 3 x 3 x 3 time

1

1

Signature Sequence

Exchanged Spreading

x0 x1

1

1

×

x 2 x 3 x 0 x 1 x 2 x 3 x0 x1 x 2 x 3 x 0 x 1 x 2 x 3

Data Sequence

x0 x1

x 2 x 3 x 0 x 1 x 2 x 3 x0 x1 x 2 x 3 x 0 x 1 x 2 x 3 time

Figure 3.11: Spreading with the roles of data sequence and signature sequence exchanged for spreading signature of {1, 1, 1, 1} with a data block size of 4. •

Both spread narrow-band data into broader band.



They achieve processing gain or spreading gain from spreading.



They both maintain low PAPR because of the single carrier transmission.

An interesting relationship between orthogonal DS-CDMA and IFDMA is that by exchanging the roles of spreading sequence and data sequence, DS-CDMA modulation becomes IFDMA modulation [30], [31]. An example of this observation is illustrated in Figure

40

3.11. We could see that the result of the spreading with exchanged roles is in the form of IFDMA modulated symbols in Figure 3.7. One advantage of SC-FDMA over DS-CDMA/FDE is that channel dependent resource scheduling is possible to exploit frequency selectivity of the channel.

3.6.

SC-FDMA Implementation in 3GPP LTE Uplink

In this section, we describe the physical layer implementation of SC-FDMA in 3GPP LTE frequency division duplex (FDD) uplink according to [10]. 3.6.1. Modulation and Channel Coding Available data modulation schemes are π/2-offset BPSK, QPSK, 8-PSK, and 16-QAM. Turbo code based on 3GPP UTRA Release 6 is used for forward error correcting (FEC) code. 3.6.2. Sub-frame Structure In 3GPP LTE, the basic unit of a transmission is a sub-frame. Figure 3.12 shows the basic sub-frame structure in the time domain.

SB #2

CP

LB #5

CP

LB #4

CP

LB #3

CP

LB #2

CP

SB #1

CP

LB #1

CP

CP

1 sub-frame = 0.5 ms

LB #6

Figure 3.12: Basic sub-frame structure in the time domain. A sub-frame which has duration of 0.5 ms consists of six long blocks (LB) and two short blocks (SB). Cyclic prefix (CP) is added in front of each block. Long blocks are used for control and/or data transmission and short blocks are used for reference (pilot) signals for coherent demodulation and/or control/data transmission. Both localized and distributed

41

subcarrier mapped data use the same sub-frame structure. The minimum transmission time interval (TTI) for uplink is equal to the sub-frame duration, 0.5 ms. It is possible to concatenate multiple sub-frames into longer uplink TTI’s and the TTI configuration can be either semi-static or dynamic in such case. In 3GPP LTE uplink, subcarriers in the guard band area are intentionally set to zero amplitude to maintain similar subcarrier structure as that of downlink OFDMA, which is illustrated in Figure 3.13.

Total Bandwidth (M subcarriers)

Occupied Subcarriers

fc

Frequency (Subcarrier)

Figure 3.13: Physical mapping of a block in RF frequency domain (fc: carrier center frequency). Figure 3.14 illustrates the generation of a block and Table 3.1 shows the numerology for different spectrum allocations.

{ x0 , x1 …, xN −1}

Serial-toParallel

subc arrier

Zeros

M-1

Subcarrier Mapping

Npoint DFT

0

Zeros

Figure 3.14: Generation of a block.

Mpoint IDFT

Parallel-toSerial

{ xɶ0 , xɶ1 …, xɶM −1}

42

Table 3.1: Parameters for uplink SC-FDMA transmission scheme in 3GPP LTE. Bandwidth (MHz)

Sub-frame duration (ms)

LB size (µs/# of occupied subcarriers /FFT size)

SB size (µs/# of occupied subcarriers /FFT size)

20

0.5

66.67/1200/2048

33.33/600/1024

(4.13/127) or (4.39/135)

15

0.5

66.67/900/1536

33.33/450/768

(4.12/95) or (4.47/103)

10

0.5

66.67/600/1024

33.33/300/512

5

0.5

66.67/300/512

33.33/150/256

2.5

0.5

66.67/150/256

33.33/75/128

1.25

0.5

66.67/75/128

33.33/38/64

CP duration (µs/# of subcarriers)

(4.1/63) or (4.62/71) (4.04/31) or (5.08/39) (3.91/15) or (5.99/23) (3.65/7) or (7.81/15)

Note that the first CP in the sub-frame has longer duration to include ramp up and ramp down time. For data and control transmission, we can organize the subcarrier resources in the frequency domain into a number of resource units (RU). In other words, we can assign the subcarriers in groups or chunks of certain number rather than individually. Each RU consists of a number of localized or distributed subcarriers within a long block. Table 3.2 gives the number of RU’s and the number of subcarriers per RU for different bandwidth allocations. The number of subcarriers per RU may change in the future based on the outcome of the current interference coordination study. One or more RU’s can be assigned to a user. When

43

more than one localized RU’s are assigned to a user, they should be contiguous in frequency domain and when more than one distributed RU’s are assigned, the subcarriers assigned should be equally spaced. Table 3.2: Number of RU’s and number of subcarrriers per RU for LB. Bandwidth (MHz)

1.25

2.5

5.0

10.0

15.0

20.0

Number of available subcarriers

75

150

300

600

900

1200

Number of available RU’s

3

6

12

24

36

48

Number of subcarriers per RU

25

25

25

25

25

25

3.6.3. Reference (Pilot) Signal Structure Uplink reference signals are transmitted within the two short blocks as mentioned in 3.6.2. They are received and used at the base station for uplink channel estimation in coherent demodulation/detection and for possible uplink channel quality estimation for uplink channel dependent scheduling. We should note that the bandwidth of each subcarrier for the reference signal is twice that for the data signal since the short block has half the length of a long block. We can construct the reference signal using CAZAC (Constant Amplitude Zero AutoCorrelation) sequences such as Zadoff-Chu polyphase sequences. Zadoff-Chu CAZAC sequences are defined as

 − j 2π r  k 2 + qk  , L 2   e ak =  r k ( k +1)  − j 2π L  2 + qk  e

k = 0,1,2,⋯, L −1; for L even

(3.15) , k = 0,1,2,⋯, L −1; for L odd

44

where r is any integer relatively prime to L and q is any integer [32]. The set of Zadoff-Chu CAZAC sequences has the following properties: •

Constant amplitude.



Zero circular autocorrelation.



Flat frequency domain response.



Circular cross-correlation between two sequences is low and it has constant magnitude provided that L is a prime number.

Orthogonality among uplink reference signals can be achieved using the following methods: •

Frequency division multiplexing (FDM): Each uplink reference signal is transmitted across a distinct set of subcarriers. This solution achieves signal orthogonality in the frequency domain.



Code division multiplexing (CDM): Reference signals are constructed that are orthogonal in the code domain with the signals transmitted across a common set of sub-carriers. As an example, individual reference signals may be distinguished by a specific cyclic shift of a single CAZAC sequence.



Time division multiplexing (TDM): Each user will transmit reference signal in different short blocks. Orthogonality is achieved in the time domain.



A combination of the methods above.

Figure 3.15 illustrates an example of FDM and CDM pilots.

45 User 1 User 2 User 3 subcarriers FDM Pilots

subcarriers CDM Pilots

Figure 3.15: FDM and CDM pilots for three simultaneous users with 12 total subcarriers.

3.7.

Summary and Conclusions

Single carrier FDMA (SC-FDMA) is a new multiple access scheme that is currently being adopted in 3GPP LTE uplink. SC-FDMA utilizes single carrier modulation and frequency domain equalization, and has similar performance and essentially the same overall complexity as those of OFDMA system. A salient advantage over OFDMA is that the SC-FDMA signal has lower PAPR because of its inherent single carrier transmission structure. SC-FDMA has drawn great attention as an attractive alternative to OFDMA, especially in the uplink communications where lower PAPR greatly benefits the mobile terminal in terms of transmit power efficiency and manufacturing cost. SC-FDMA has two different subcarrier mapping schemes; distributed and localized. In distributed subcarrier mapping scheme, user’s data occupy a set of distributed subcarriers and we achieve frequency diversity. In localized subcarrier mapping scheme, user’s data inhabit a set of consecutive localized subcarriers and we achieve frequency-selective gain through channel-dependent scheduling (CDS). The two flavors of subcarrier mapping schemes give the system designer a flexibility to adapt to the specific needs and requirements.

46

SC-FDMA also bears similarities to direct sequence CDMA (DS-CDMA) with FDE in terms of; bandwidth spreading, processing gain from spreading, and low PAPR because of the single carrier transmission. An advantage of SC-FDMA over DS-CDMA/FDE is the ability to utilize channel dependent resource scheduling to exploit frequency selectivity of the channel.

Chapter 4

MIMO SCSC-FDMA

Multiple input multiple output (MIMO) techniques gathered much attention in recent years as a way to drastically improve the performance of wireless mobile communications. MIMO technique utilizes multiple antenna elements on the transmitter and the receiver to improve communication link quality and/or communication capacity [33]. A MIMO system can provide two types of gain; spatial diversity gain and spatial multiplexing gain [34]. Spatial diversity improves the reliability of communication in fading channels and spatial multiplexing increases the capacity by sending multiple streams of data in parallel through multiple spatial channels. Transmit eigen-beamforming (TxBF) with unitary precoding is a spatial multiplexing technique that utilizes the eigen-structure of the channel to generate independent spatial channels. Practical considerations of TxBF that affect the performance and the overhead of the system are precoder quantization/averaging and feedback delay. In this chapter, we first give a general overview of MIMO concepts. In section 4.2, we describe the parallel decomposition of a MIMO channel for narrowband and wideband transmission. We also illustrate the realization of MIMO spatial multiplexing in SC-FDMA. In section 4.3, we introduce the SC-FDMA TxBF with unitary precoding technique and

47

48

numerically analyze the link level performance with practical considerations.

4.1.

Spatial Diversity and Spatial Multiplexing in MIMO Systems

The basic idea behind spatial diversity techniques is to combat channel fading by having multiple copies of the transmitted signal go through independently fading propagation paths. At the receiver, multiple independently faded replicas of the transmitted data signal are coherently combined to achieve a more reliable reception. Spatial diversity gain is the SNR exponent of the error probability and intuitively, it corresponds to the number of independently faded paths. It is well known that in a system with Nt transmit antennas and Nr receive antennas, the maximal diversity gain is Nt⋅Nr assuming the path gains between the individual transmit and receive antenna pairs are independently and identically distributed (i.i.d.) Rayleigh-faded [35]. Smart antenna techniques [36], Alamouti transmit diversity scheme [37], and space-time coding [38] are some of well known spatial diversity techniques. If spatial diversity is a means to combat fading, spatial multiplexing is a way to exploit fading to increase the data throughput. In essence, if the path gains among individual transmitreceive antenna pairs fade independently such that the channel matrix is well-conditioned, multiple parallel spatial channels can be created and data rate can be increased by transmitting multiple streams of data through the spatial channels. Spatial multiplexing is especially important in the high-SNR regime where the system is degree-of-freedom limited as opposed to power limited in the low-SNR regime. A scheme achieves a spatial multiplexing gain of r if the supported data rate approaches r⋅log(SNR) (bits/s/Hz) and we can think of the multiplexing gaing as the total number of degrees of freedom to communicate. In a system with Nt transmit

49

antennas and Nr receive antennas for high SNR, Foschini [39] and Telatar [40] showed that the capacity of the channel increases linearly with m = min(Nt, Nr) if the channel gains among the antenna pairs are i.i.d. Rayleigh-faded. Thus there is m number of parallel channels or m number of degrees of freedom. BLAST (Bell Labs Space-Time Architecture) [39] is a typical spatial multiplexing technique. For a given MIMO channel, both diversity and multiplexing gains can be achieved simultaneously but there is a fundamental trade-off between the two gains. For example, Zheng and Tse [41] showed that the optimal diversity gain achievable by any coding scheme of block length larger than Nt + Nr − 1 with multiplexing gain m (integer) is precisely (Nt − m)⋅( Nr − m) for i.i.d. Rayleigh slowly-fading channel with Nt transmit and Nr receive antennas. This implies that out of the total resource of Nt transmit and Nr receive antennas, it is as though m transmit and m receive antennas were used for multiplexing and the remaining Nt − m transmit and Nr − m receive antennas provided the diversity. In summary, higher spatial diversity gain comes at the

price of lowering spatial multiplexing gain and vice versa.

4.2.

MIMO Channel

First, we consider the narrowband MIMO channel. A point-to-point narrowband MIMO channel with Nt transmit antennas and Nr receive antennas is illustrated in Figure 4.1. We can represent such a MIMO system by the following discrete-time model.

50

h11 hNr 1

x1

y1

h21

y2

x2

x Nt

hNr Nt yNr

 h11 ⋯ h1Nt  ⋮ H = ⋮ ⋱ h  N r 1 ⋯ hNr Nt

    

Figure 4.1: Description of a MIMO channel with Nt transmit antennas and Nr receive antennas.  y1   h11 ⋯ h1Nt   x1   n1          ⋮ ⋅ ⋮  +  ⋮   ⋮ = ⋮ ⋱  y  h ⋯ hN r Nt   xNt   nNr  Nr  Nr 1    

  =y

=H

=x

(4.1)

=n

⇒ y = H ⋅x+n

where x is the Nt ×1 transmitted signal vector, y is the Nr ×1 received signal vector, n is the Nr ×1 zero-mean complex Gaussian noise, and H is the Nr×Nt complex matrix of channel gains hij representing the gain from jth transmit antenna to ith receive antenna. By singular value decomposition (SVD) theorem, channel matrix H can be decomposed as follows.

51

H = UDV H

(4.2)

where U is an Nr×Nr unitary matrix, V is an Nt×Nt unitary matrix, D is an Nr×Nt non-negative diagonal matrix, and (i) H is a Hermitian (conjugate transpose) operation. The diagonal entries of D are the non-negative square roots of the eigenvalues of HH H , the columns of U are the eigenvectors of HH H , and the columns of V are the eigenvectors of H H H . We can rewrite (4.1) using (4.2) as

y = UDV H x + n

(4.3)

Multiplying U H on both sides of (4.3), it becomes H UH y =U U DV H x + U H n  =I

U y = DV x + U n H

H

(4.4)

H

Let yɶ = U H y , xɶ = V H x , and nɶ = U H n , then, yɶ = Dxɶ + nɶ

(4.5)

where nɶ has the same statistical properties as n since U is a unitary matrix. Thus nɶ is an Nr ×1 zero-mean complex Gaussian noise.

We can see from (4.5) that the MIMO transmission can be decomposed into up to m = min(Nt, Nr) parallel independent transmissions, which is the basis for the spatial multiplexing

gain. For a wideband MIMO channel, the entire band can be subdivided into subbands, or subcarriers. Then, the MIMO channel for each subcarrier becomes a narrowband MIMO channel. Let the entire band be subdivided in to M subcarriers, then for kth subcarrier

52

( 0 ≤ k ≤ M − 1 ),

 Y1, k   H11,k ⋯ H1Nt ,k   X 1,k   N1, k          ⋱ ⋮ ⋅ ⋮  +  ⋮   ⋮ = ⋮  YN , k   H N 1,k ⋯ H N N ,k   X N ,k   N N , k  r r r t t r 

  

   

  

  =Yk

=Hk

= Xk

(4.6)

= Nk

⇒ Yk = H k ⋅ X k + N k where X k is the Nt×1 transmitted signal vector, Yk is the Nr×1 received signal vector, N k is the Nr×1 zero-mean complex Gaussian noise, and Hk is the Nr×Nt complex matrix of channel gains Hij,k representing the gain from jth transmit antenna to ith receive antenna. Similarly with a narrowband MIMO channel, Hk can be decomposed into U k DkVkH and (4.6) can be expressed as Yɶk = Dk Xɶ k + Nɶ k

(4.7)

where Yɶk = U k H Yk , Xɶ k = Vk H X k , and Nɶ k = U k H N k . We can use spatial multiplexing MIMO techniques in SC-FDMA by applying the spatial processing on a subcarrier-by-subcarrier basis, somewhat similar to MIMO-OFDM [42], in the frequency domain after DFT. A block diagram of a spatial multiplexing MIMO SC-FDMA system is shown in Figure 4.2.

N-point DFT

N-point DFT

Spatial Mapping

53 Subcarrier Mapping

M-point IDFT

Add CP / PS

DA C / RF

Subcarrier Mapping

M-point IDFT

Add CP / PS

DA C / RF

Detect

N-point IDFT

Detect

N-point IDFT

Spatial Combining / Equalization

MIMO Channel

Subcarrier De-mapping

M-point DFT

Remove CP

RF / A DC

Subcarrier De-mapping

M-point DFT

Remove CP

RF / A DC

Figure 4.2: Block diagram of a spatial multiplexing MIMO SC-FDMA system.

4.3.

SC-FDMA Transmit Eigen-Beamforming with Unitary Precoding

Transmit eigen-beamforming (TxBF) with unitary precoding utilizes the eigen-structure of the channel to achieve spatial multiplexing [43], [44]. Figure 4.3 illustrates a simpified block diagram of a unitary precoded TxBF SC-FDMA MIMO system. We describe below the basic processes of unitary precoded TxBF system. We assume the channel is time-invariant and the channel estimation is perfect, and we follow the notations in (4.6). First, the MIMO channel matrix Hk for subcarrier k is decomposed as follows using SVD for each subcarrier at the receiver. H k = U k DkVkH

(4.8)

54

Xk

Unitary Precoding

Xɶ k = Vˆk X k MIMO Channel Hk

Hk X k

⊕ Nk

Vˆk = F (Vk )

Yk

Receiver (LMMSE)

Zk

Feedback Processing: F (⋅) (Av eraging & quantization of V k’s)

Figure 4.3: Simpified block diagram of a unitary precoded TxBF SC-FDMA MIMO system. We feedback Vk to the transmitter and use it as the precoding matrix for transmission. We may quantize and average the Vk’s at the receiver to reduce feedback overhead. We denote the quantization and averaging processes as F(⋅) and Vˆk = F (Vk ) . The quantized and averaged Vˆk ’s are then sent to the transmitter via the feedback channel. The transmitter precodes the data signal X k with Vˆk as follows.

Xɶ k = Vˆk X k

(4.9)

After the transmitted signal propagates through the MIMO channel Hk, the received signal Yk is represented as follows. Yk = H k Xɶ k + N k

(4.10)

At the receiver, we perform linear MMSE (LMMSE) estimation for joint equalization and MIMO detection. Let Z k = Ak Yk where Ak is an Nt ×Nr complex matrix. Then, our goal is to

55

{

find Ak such that mean square error between X k and Z k ( E X k − Z k

2

} ) is minimum. By

applying orthogonality principal, we obtain the following solution for Ak .

(

Ak = RXX ,k Hˆ kH Hˆ k RXX ,k Hˆ kH + RNN ,k

)

−1

(4.11)

where Hˆ k = H kVˆk , RXX ,k = X k X kH , and RNN ,k = N k N kH . There are two practical factors that impact the performance of precoded TxBF; precoder quanization/averaging and feedback delay. Since the receiver feedbacks the precoding matrix to the transmitter, the receiver often quantizes and also averages across all subcarriers the precoding matrix to reduce feedback overhead. We could expect some degradation in performance because we lose some of precoder information during quantization and averaging processes. Feedback delay becomes a major factor in the performance of precoded TxBF when the channel is changing very fast since the precoder is extracted directly from the current channel matrix. In the following sections, we give numerical analysis for a 2x2 uplink MIMO system to quantify the impact of imperfect feedback on the link level performance. We base the simulation setup on 3GPP LTE uplink and we use the following simulation parameters and assumptions. •

Carrier frequency: 2.0 GHz.



Transmission bandwidth: 5 MHz.



Symbol rate: 7.68 million symbols/sec.



TTI length: 0.5 ms.



Number of blocks per TTI: 6 long blocks (LB).

56



Number of occupied subcarriers per LB: 128.



FFT block size: 512.



Cyclic Prefix (CP) length: 32 symbols.



Subcarrier mapping: Distributed.



Pulse shaping: Time domain squared-root raised-cosine filter (rolloff factor = 0.22) with 2x oversampling.



Channel model: 3GPP SCME C quasi-static Rayleigh fading with mobile speed of 3 km/h [45]. We assume the channel is constant during the duration of a block.



Antenna configurations: 2 transmit and 2 receive antennas



Data modulation: 16-QAM for 1st stream and QPSK for 2nd stream.



Channel coding: 1/3-rate turbo code.



We use linear MMSE equalizer described in (4.11).



We assume no feedback error and perfect channel estimation at the receiver.

4.3.1. Impact of Imperfect Feedback: Precoder Quantization/Averaging We directly quantize the 2x2 precoding matrix Vk in the form of Jacobi rotation matrix which is a unitary matrix [46]. We use the following matrix to quantize Vk.  cos θˆk ˆ Vk =  ˆ sin θˆk ⋅ e jφk

ˆ − sin θˆk ⋅ e jφk   cos θˆk 

(4.12)

where for Bamp bits to quantize the amplitude component,  

θˆk ∈ ( 2l − 1)

π 4⋅2

Bamp

; l = 1,⋯ , 2

Bamp

  

(4.13)

57 4

0.8 Output phase [rad]

Output amplitude [linear]

1

0.6 0.4 0.2 0 0

0.2 0.4 0.6 0.8 Input amplitude [linear]

1

(a)

2

0

-2

-4 -4

-2 0 2 Input phase [rad]

4

(b)

Figure 4.4: Input-output characteristics of the quantizers; (a) Amplitude quantizer (1 bit); (b) Phase quantizer (2 bits). and for Bphase bits to quantize the phase component,  2π l −π B  2 phase

φˆk ∈ 

; l = 1,⋯ , 2

B phase

 . 

(4.14)

Figure 4.4 shows the input-output quantization characteristics. For phase quantization, we use the first antenna as a reference and normalize the phase of the second antenna element with regards to that of first antenna element. To reduce the overhead of the feedback, {Vk}’s are averaged over multiple continuous subcarriers. Figure 4.5 shows the Monte Carlo simulation results for frame error rate (FER) performance of the 2x2 SC-FDMA unitary precoded TxBF system. We can see that the performance loss at 1% FER with regards to no averaging and no quantization case is 1.3 dB for the worst case (25-subcarrier averaging and 2-bit phase-only quantization).

58 FER, 2x2 TxBF, SCME C 3km/h, 16QAM-QPSK 1/3, Ideal ChEst, No Feedback Delay, Jacobi Direct Quant.

0

10

-1

FER

10

-2

10

-3

10

-4

10

0

No avr., no quant. No avr., phase-only(2 bits) quant. No avr., amp.(1 bit)-phase(2 bits) quant. 25 SCs avr., phase-only(2 bits) quant. 25 SCs avr., amp.(1 bit)-phase(2 bits) quant. 12 SCs avr., phase-only(2 bits) quant. 12 SCs avr., amp.(1 bit)-phase(2 bits) quant. 1

2

3

4

5

6 7 SNR [dB]

8

9

10

11

12

Figure 4.5: FER performance of a 2x2 SC-FDMA unitary precoded TxBF system with feedback averaging and quantization. Figure 4.6 (a) shows the effect of different quantization bit resolutions (no averaging) and Figure 4.6 (b) shows the effect of different averaging sizes (1-bit amplitude and 2-bit phase quantization). As we use more bits and smaller averaging size, we see less performance degradation. Table 4.1 summarizes the tradeoff between feedback overhead and performance loss based on the results in Figure 4.5. Performance loss is the SNR level increase at 1% FER point with regards to that of no averaging and no quantization result. As we increase the feedback overhead,

59 FER, 2x2 TxBF, SCME C 3km/h, 16QAM-QPSK 1/3, Ideal ChEst, No Feedback Delay, Jacobi Direct Quant., No avr. 0

10

-1

FER

10

-2

10

-3

10

No quant. Phase-only (2 bits) quant. Amp. (1 bit)-phase (2 bits) quant. -4

10

0

1

2

3

4

5 6 7 8 SNR per Antenna [dB]

9

10

11

12

(a) FER, 2x2 TxBF, SCME C 3km/h, 16QA M-QPSK 1/3, Ideal ChEst, No Feedback Delay, Jacobi Direct Quant.

0

10

-1

FER

10

-2

10

No avr., no quant. 12 SCs avr., quant. 25 SCs avr., quant. 50 SCs avr., quant. 100 SCs avr., quant.

-3

10

-4

10

0

* 1 bit amp. / 2 bit phase quantization 1

2

3

4

5 6 7 8 SNR per Antenna [dB]

9

10

11

12

(b) Figure 4.6: FER performance; (a) with different quantization bit resolutions (no averaging); (b) with different averaging sizes (1-bit amplitude and 2-bit phase quantization).

60

that is, increase the amount of precoding information, there is less performance loss. Table 4.1. Summary of feedback overhead vs. performance loss. Averaging size # of (subcarriers per RU) RU’s

Quantization resolution (bits)

Feedback overhead

Performanc e loss

25

12

Phase: 2, Amp.: none

2 bits × 12 = 24 bits

1.3 dB

25

12

Phase: 2, Amp.: 1

3 bits × 12 = 36 bits

0.6 dB

12

25

Phase: 2, Amp.: none

2 bits × 25 = 50 bits

1 dB

12

25

Phase: 2, Amp.: 1

3 bits × 25 = 75 bits

0.5 dB

4.3.2. Impact of Imperfect Feedback: Feedback Delay Delay is inevitable in any realistic feedback system. When the channel is changing fast due to the user’s motion or change of the environment, the precoder may be outdated and it will affect the receiver performance. Figure 4.7 shows the impact of feedback delay for different user speeds. We consider feedback delays of 2, 4, and 6 TTI’s with no feedback quantization or averaging. We can see from Figure 4.7 (a) that at low speed of 3 km/h, there is only a trivial loss of performance even for the longest delay of 6 TTI’s. But as the user speed becomes faster, we can see the performance degradation become more severe as illustrated in Figure 4.7 (b). Clearly, in high mobility and long feedback delay situations, TxBF with unitary precoding does not perform well at all.

4.4.

Summary and Conclusions

MIMO technique utilizes multiple antenna elements on the transmitter and the receiver to improve communication link quality and/or communication capacity and it has two aspects in

61 FER, 2x2 TxBF, SCME C 3km/h, 16QAM-QPSK 1/3, Ideal ChEst, No Quant.

FER

10

10

10

10

FER, 2x2 TxBF, SCME C 150km/h, 16QAM-QPSK 1/3, Ideal ChEst, No Quant.

0

10

-1

10

FER

10

-2

-3

No delay 2 TTI delay 4 TTI delay 6 TTI delay

10

-4

0

2

10

4 6 SNR per Antenna [dB]

(a)

8

10

10

0

-1

-2

-3

No delay 2 TTI delay 4 TTI delay 6 TTI delay

-4

0

5

10 15 SNR per Antenna [dB]

20

(b)

Figure 4.7: FER performance of a 2x2 SC-FDMA unitary precoded TxBF system with feedback delays of 2, 4, and 6 TTI’s; (a) user speed of 3 km/h; (b) user speed of 150 km/h. terms of performance improvement; spatial diversity and spatial multiplexing. Spatial diversity improves the reliability of communication in fading channels and spatial multiplexing increases the capacity by sending multiple streams of data in parallel through multiple spatial channels. Transmit eigen-beamforming (TxBF) with unitary precoding is a spatial multiplexing technique that utilizes the eigen-structure of the channel to generate independent spatial channels. Practical considerations of TxBF that affect the performance and the overhead of the system are precoder quantization/averaging and feedback delay. Quantizing and averaging the precoding matrix reduces the feedback overhead but the link performance suffers. In this chapter, we gave an overview of the realization of MIMO spatial multiplexing in SCFDMA. We specifically analyzed the unitary precoded TxBF technique on an SC-FDMA system.

62

We showed with numerical simulations that there is a trade-off between the feedback overhead and the link performance loss. We also showed that feedback delay is another impairment that degrades the performance especially for the high speed mobile users.

Chapter 5

Peak Power Characteristics Characteristics of an SCSC-FDMA Signal: Analytical Analysis

In this chapter, we analytically characterize the peak power of SC-FDMA modulated signals using the Chernoff bound. We will analyze the peak power characteristics numerically in chapter 6. In [47], Wulich and Goldfeld showed that the amplitude of a single carrier (SC) modulated signal does not have a Gaussian distribution and that it is difficult to analytically derive the exact form of the distribution. As an alternative, they explained a way to derive an upper bound for the complementary distribution of the instantaneous power using the Chernoff bound. We follow their derivation and apply it to SC-FDMA modulated signals to characterize the peak power analytically. 2

The peak power of any signal x(t) is the maximum of its squared envelope x (t ) . However, for a continuous random process, max x (t ) random process with discrete values where max x (t )

2

2

could be unbounded. Even for

is bounded, it may occur at very low

probability which is not very useful in practice. Rather, the distribution of x (t )

2

is more

useful and we describe it with the complementary cut-off probability. For a given w, cut-off

63

64

{

}

probability is defined as Pr x (t ) ≤ w = F x ( t ) distribution function (CDF) of

{

}

2

( w ) , where

F x ( t ) 2 ( w ) is the cumulative

2

x (t ) , and the complementary cut-off probability is

{

}

Pr x (t ) ≥ w = 1 − F x ( t ) 2 ( w ) . Pr x (t ) ≥ w 2

2

2

is also referred to as complementary

cumulative distribution function (CCDF). In this chapter, we first derive an upper-bound of the CCDF of the instantaneous power for interleaved FDMA (IFDMA) with pulse shaping and show the bounds for BPSK and QPSK with raised-cosine pulse shaping filter. In section 5.2, we show the modified upperbound of the CCDF of the instantaneous power for localized FDMA (LFDMA) without pulse shaping. In section 5.3, we compare the results of sections 5.1 and 5.2 with the analytical CCDF of PAPR for OFDM.

5.1.

Upper Bound for IFDMA with Pulse Shaping

In chapter 3 section 3.3.1, we derived the time domain representation of the IFDMA symbols and showed that it is a repetition of the original input symbols with a scaling factor of 1/Q. Since the IFDMA signal has the same structure as the conventional SC modulation signal, we first derive the upper bound on the CCDF of the instantaneous power of a conventional SC modulated signal. We consider a baseband representation of the conventional SC modulated signal

x(t , s ) =



∑s

k =−∞

where

{sk }k =−∞ ∞

k

p (t − kT )

(5.1)

are transmitted symbols, s = [… , s−1 , s0 , s1 ,…] , p(t) is the pulse shaping

65

filter, and T is the symbol duration. sk belongs to a modulation constellation set C of size B, that is, sk ∈ C = {cb : 0 ≤ b ≤ B − 1} and it is uniformly distributed. We assume

{sk } ’s

are

independent of each other. Let us define a random variable Z for a given t0 ∈ [0, T) as follows. Z ≜ x(t0 , s ) =



∑s

k =−∞

where ak = sk p (t0 − kT ) .

k

p (t0 − kT ) =

{ak } ’s



∑a

k =−∞

(5.2)

k

are also mutually independent since

{sk } ’s

are mutually

independent. Since sk is uniformly distributed over C and from (5.2), Pr [ ak = cb p (t0 − kT )] = Pr [ sk = cb ] =

1 B

(5.3)

where 0 ≤ b ≤ B − 1 . Our goal is to characterize the following CCDF. 2 2 Pr  x(t0 , s ) ≥ w = Pr  Z ≥ w = Pr  Z ≥ δ     

(5.4)

where w ≥ 0 and δ = w . Let us assume the modulation constellation set C is real and its constellation points are symmetric. Then, Pr  Z ≥ δ  = Pr [ Z ≥ δ ] + Pr [ Z ≤ −δ ] = 2 Pr [ Z ≥ δ ]

Using the Chernoff bound, the following inequality holds [48], [49].

(5.5)

66

{ }

Pr [ Z ≥ δ ] ≤ e−νδˆ E eνˆZ

(5.6)

where νˆ is a solution of the following equation.

{

}

{ }

E Zeν Z − δ ⋅ E eν Z = 0

(5.7)

By expanding E {Zeν Z } and E {eν Z } , (5.7) becomes ∞



k =−∞

{ } =δ E {e }

E ak eνˆak

(5.8)

νˆak

for ν = νˆ . By solving (5.8), we obtain νˆ for a given δ. Since (5.8) is not in a closed form, we evaluate the solution numerically. 2 We can upper-bound the CCDF Pr  x(t0 , s ) ≥ w as follows from (5.4), (5.5), and  

(5.6). ∞

{ }

2 ˆ Pr  x (t0 , s ) ≥ w ≤ 2e −νδˆ ∏ E eν ak ≜ Pub , SC   k =−∞

(5.9)

The details of the derivations for (5.8) and (5.9) are in section A.1 of appendix A. Since it is impossible to consider the infinite span in (5.8) and (5.9), we limit the span by only considering -Kmax ≤ k ≤ Kmax. As long as the IFDMA input block size N is larger than Kmax, the instantaneous power distribution of the IFDMA signal will have the same upper

bound as the conventional SC signal since the input symbols are mutually independent within the span of -Kmax ≤ k ≤ Kmax. In the next subsections, we derive the upper bounds specifically for BPSK and QPSK

67

modulations. We consider the following raised-cosine pulse.

sin (π t / T ) cos (πα t / T ) ⋅ πt /T 1 − 4α 2t 2 / T 2

p(t ) =

(5.10)

where 0 ≤ α ≤ 1 is the roll-off factor. 5.1.1. Upper Bound for BPSK For BPSK, we use the following constellation set.

CBPSK = {−1,1}

(5.11)

Then, (5.8) becomes

K max



k =− K max

1 1 ˆ ˆ p (t0 − kT )eν p (t0 − kT ) − p (t0 − kT )e −ν p ( t0 − kT ) 2 2 =δ 1 νˆ p (t0 − kT ) 1 −νˆ p (t0 − kT ) e + e 2 2

(5.12)

After we determine νˆ from (5.12), the upper bound in (5.9) becomes

( BPSK ) ub , SC

P

≜e

ˆ −νδ

1   2

2 K max

∏ ( eν K max

ˆ p ( t0 − kT )

k =− K max

+ e −ν p ( t0 − kT ) ˆ

)

(5.13)

and we upper-bound the CCDF as 2 ) Pr  x ( BPSK ) (t0 , s ) ≥ w ≤ Pub( BPSK , SC  

(5.14)

The details of the derivations for (5.12) and (5.13) are in section A.2 of appendix A. ) Figure 5.1 shows Pub( BPSK for Kmax= 8 , T = 1, and t0 = T/2 = 0.5 along with the , SC

empirical results using Monte Carlo simulation. We considered roll-off factor α of 0, 0.2, 0.4,

68

10

Pr(|Z|2 > w)

10

10

10

10

CCDF of Instantaneous Power for IFDMA (BPSK)

0

-1

-2

-3

α = 0.6

α = 0.4

-4

α = 0.2

0

2

4 w [dB]

α=0 6

8

Figure 5.1: CCDF of instantaneous power for IFDMA with BPSK modulation and different values of roll-off factor α. Dotted lines are empirical results and solid lines are the upper bounds. and 0.6. For the Monte Carlo simulation, we generated 1000 random IFDMA-modulated blocks with total number of symbols M = 256, number of input symbols N = 64, and spreading factor Q = 4, and produced a histogram. We can see that the upper bound we derived in (5.13) is valid compared to the empirical results and that the bound is rather tight in the tail region of the distribution which we are most interested in. Another interesting observation is the fact that the peak power at a given probability increases as the roll-off factor of the raised-cosine filter becomes closer to zero. 5.1.2. Modified Upper Bound for QPSK ) Figure 5.2 shows the modified upper-bound for QPSK, Pub(QPSK , for Kmax= 8 , T = 1, and , SC

69 10

Pr(|Z|2 > w)

10

10

10

CCDF of Instantaneous Power of IFDMA (QPSK)

0

-1

-2

α = 0.2 -3

α=0 α = 0.6

10

-4

α = 0.4

0

2

4

6

8

10

w [dB]

Figure 5.2: CCDF of instantaneous power for IFDMA with QPSK modulation and different values of roll-off factor α. Dotted lines are empirical results and solid lines are the upper bounds. t0 = T/2 = 0.5 along with the empirical results using Monte Carlo simulation. The derivation ) of Pub(QPSK is in section A.3 of appendix A. We considered roll-off factor α of 0, 0.2, 0.4, , SC

and 0.6. For the Monte Carlo simulation, we generated 1000 random IFDMA-modulated blocks with total number of symbols M = 256, number of input symbols N = 64, and spreading factor Q = 4, and produced a histogram. To make the upper bound tighter, we used

γ = 2 (see section A.3 of appendix A). We can see that the modified upper bound we derived is valid compared to the empirical results. As with the case of BPSK in section 5.1.1, the peak power at a given probability increases as the roll-off factor of the raised-cosine pulse

70

10

Pr(|Z|2>w)

10

10

10

CCDF of Instantaneous Power for LFDMA (BPSK)

0

-1

-2

N=4

-3

N=8

N =32 10

-4

0

2

4

6

8

10

w [dB]

Figure 5.3: CCDF of instantaneous power for LFDMA with BPSK modulation and different values of input block size N. Dotted lines are empirical results and solid lines are the upper bounds. shaping filter becomes closer to zero.

5.2.

Modified Upper Bound for LFDMA and DFDMA

In chapter 3 sections 3.3.2 and 3.3.3, we showed that the time domain symbols of localized FDMA (LFDMA) and distributed FDMA (DFDMA) signals have the same structure. Thus the distributions of the instantaneous power are the same for both signals. We do not consider pulse shaping in the following analysis because of the complexity of the derivation. Figure 5.3 shows the modified upper bound for BPSK, Pub,LFDMA, for M = 256 and different values of N and Q, along with the empirical results using Monte Carlo simulation. The details of the derivation for Pub,LFDMA is in section A.4 of appendix A. For the Monte

71

Carlo simulation, we generated 1000 random LFDMA-modulated blocks and produced a histogram. To make the upper bound tighter, we used γ =

π 4

(see section A.4 of appendix

A). We can see that the upper bound we derived is valid compared to the empirical results. We can also observe that the peak power at a given probability increases as the input block size block N increases.

5.3.

Comparison with OFDM

We can express the CCDF of the PAPR of an OFDM data block for N subcarriers with Nyquist rate sampling as follows [50], [51].

(

Pr {PAPR ≥ w} = 1 − 1 − e − w

)

N

(5.15)

Let us assume the input symbols have unit power. Then we can use (5.15) and compare it with the CCDF of the instantaneous power of IFDMA and that of LFDMA. Figure 5.4 compares the analytical upper bounds of CCDF for IFDMA and LFDMA with the theoretical CCDF for OFDM with the same block size. For IFDMA, we consider roll-off factor of 0.2. For LFDMA and OFDM, the number of occupied subcarriers N is 32 and we do not apply pulse shaping filter. We consider the case of BPSK modulation. We can see that SC-FDMA signals for both subcarrier mapping modes indeed have lower peak power at a given probability than that of OFDM. Comparing the 99.9-percentile instantaneous power

{

}

(w such that Pr Z ≥ w = 10−3 ), we see that the peak power of IFDMA is 4 dB lower than 2

that of OFDM and the peak power of LFDMA is 2.2 dB lower than that of OFDM. We can

72 10

Pr(|Z|2 > w)

10

10

10

CCDF of Instantaneous Power

0

-1

OFDM

-2

-3

IFDMA 10

-4

LFDMA

0

2

4

6 w [dB]

8

10

12

Figure 5.4: CCDF of instantaneous power for IFDMA, LFDMA, and OFDM. For IFDMA, we consider roll-off factor of 0.2. For LFDMA and OFDM, the number of occupied subcarriers N is 32 and we do not apply pulse shaping filter. We consider the case of BPSK modulation. also observe that LFDMA signal has higher peak power than IFDMA signal by 1.8 dB at 99.9percentile probability.

5.4.

Summary and Conclusion

In this chapter, we investigated the peak power characteristics for IFDMA and LFDMA signals analytically. By using Chernoff bound, we found upper bounds for both subcarrier mapping schemes and we showed results for IFDMA with BPSK and QPSK modulations and for LFDMA with BPSK modulation. We saw that the peak power of an SC-FDMA signal for both subcarrier mappings is indeed lower than that of OFDM. We also observed that IFDMA

73

has the lowest peak power among the considered subcarrier mapping schemes of SC-FDMA. With regards to the tightness of the upper bounds, the original upper bounds are rather loose except for the case of IFDMA/BPSK and we had to modify the original upper bounds to make them tighter. We prefer numerical analysis in order to precisely characterize the peak power characteristics for SC-FDMA signals, which will be the subject of the next chapter. Other interesting findings are: •

For IFDMA, the roll-off factor of the raised-cosine pulse shaping filter has a great impact on the distribution of the instantaneous power. As the roll-off factor decreases to zero, the peak power at a given probability increases.



For LFDMA, the input block size is the major factor characterizing the CCDF of peak power.

Chapter 6

Peak Power Characteristics of an SCSC-FDMA Signal: Signal: Numerical Numerical Analysis

Peak-to-average-power ratio (PAPR) is a performance measurement that is indicative of the power efficiency of the transmitter. In case of an ideal linear power amplifier where we achieve linear amplification up to the saturation point, we reach the maximum power efficiency when the amplifier is operating at the saturation point. A positive PAPR in dB means that we need a power backoff to operate in the linear region of the power amplifier. We can express the theoretical relationship between PAPR [dB] and transmit power efficiency as follows [52].

η = ηmax ⋅10



PAPR 20

(6.1)

where η is the power efficiency and ηmax is the maximum power efficiency. For class A power amplifier, ηmax is 50% and for class B, 78.5% [53]. Figure 6.1 shows the relationship graphically and it is evident from the figure that high PAPR degrades the transmit power efficiency performance. A salient advantage of SC-FDMA over OFDMA is the lower PAPR because of its inherent single carrier structure [54]. The lower PAPR is greatly beneficial in the uplink communications

74

75 100 90 80

Efficiency [%]

70 Class B 60 50 40 30 20

Class A

10 0 0

2

4

6

8

10

PAPR [dB]

Figure 6.1: A theoretical relationship between PAPR and transmit power efficiency for ideal class A and B amplifiers. where the mobile terminal is the transmitter. As we showed in section 3.3, time domain samples of the SC-FDMA modulated signals are different depending on the subcarrier mapping scheme and we can expect different PAPR characteristics for different subcarrier mapping schemes. In this chapter, we first characterize the PAPR for single antenna transmission of SCFDMA. We investigate the PAPR properties for different subcarrier mapping schemes. In section 6.2, we analyze the PAPR characteristics for multiple antenna transmission. Specifically, we numerically analyze the CCDF of PAPR for 2x2 unitary precoded TxBF SC-FDMA system described in chapter 4. In section 6.3, we propose a symbol amplitude clipping method to reduce peak power. We show the link level performance and frequency domain aspects of the proposed

76

clipping method.

6.1.

PAPR of Single Antenna Transmission Signals

In this section, we analyze the PAPR of the SC-FDMA signal for each subcarrier mapping mode. We follow the notations in Figure 3.2 of chapter 3. Let

{ xn : n = 0,1,⋯ , N − 1}

be

data

symbols

to

be

modulated.

Then,

{ X k : k = 0,1,⋯ , N − 1}

are frequency domain samples after DFT of

{ Xɶ

are frequency domain samples after subcarrier mapping, and

l

}

: l = 0,1,⋯ , M − 1

{ xɶm : m = 0,1,⋯ , M − 1}

are time symbols after IDFT of

{ Xɶ

l

{ xn : n = 0,1,⋯ , N − 1} ,

}

: l = 0,1,⋯ , M − 1 . We

represent the complex passband transmit signal of SC-FDMA x(t) for a block of data as M −1

x(t ) = e jωct ∑ xɶm p (t − mTɶ )

(6.2)

m =0

where ωc is the carrier frequency of the system, p(t) is the baseband pulse, and Tɶ is the symbol duration of the transmitted symbol xɶm . We consider raised-cosine (RC) pulse and squared-root raised-cosine (RRC) pulse which are widely used pulse shapes in wireless communications. We define the PAPR as follows for transmit signal x(t). 2

maxɶ x (t ) peak power of x(t ) PAPR = = 0≤t ≤ MTɶ 1 MT 2 average power of x(t ) x(t ) dt ∫ ɶ 0 MT

(6.3)

77

Without pulse shaping, that is, using rectangular pulse shaping, symbol rate sampling will give the same PAPR as the continuous case since SC-FDMA signal is modulated over a single carrier. Thus, we express the PAPR without pulse shaping with symbol rate sampling as follows.

PAPR =

xɶm

max

m = 0,1,⋯, M −1 M −1

1 M

∑ xɶ

m=0

2

(6.4)

2

m

Using Monte Carlo simulation, we calculate the CCDF (Complementary Cumulative Distribution Function) of PAPR, which is the probability that PAPR is higher than a certain PAPR value PAPR0 (Pr{PAPR>PAPR0}). We evaluate and compare the CCDFs of PAPR for IFDMA, DFDMA, LFDMA, and OFDMA. The following are the simulation setup and assumptions. •

We generate 104 uniformly random data blocks to acquire the CCDF of PAPR.



We consider QPSK and 16-QAM symbol constellations.



We truncate the baseband pulse p(t) from -6 Tɶ to 6 Tɶ time period and we oversample it by 8 times and we use transmission bandwidth of 5 MHz.



We consider consecutive chunks for LFDMA.



We do not apply any pulse shaping in the case OFDMA.

Figure 6.2 contains the plots of CCDF of PAPR for IFDMA, DFDMA, LFDMA, and OFDMA for total number of subcarriers M = 512, number of input symbols N = 128, IFDMA spreading factor Q = 4, and DFDMA spreading factor Qɶ = 2 . We compare the PAPR value that is exceeded with the probability less than 0.1% (Pr{PAPR>PAPR0} = 10-3), or 99.9-percentile

78 CCDF of PAPR: QPSK, Rolloff = 0.22, N = 512, N fft

0

occupied

= 128

CCDF of PAPR: 16-QAM, Rolloff = 0.22, N = 512, N fft

0

10

OFDMA

= 128

OFDMA

-1

-1

10

0

0

Pr(PAPR>PAPR )

10 Pr(PAPR>PAPR )

occupied

10

-2

10

IFDMA DFDMA

-3

10

-4

10

Dotted lines: no PS Dashed lines: RRC PS LFDMA Solid lines: RC PS

0

2

4

6 PAPR [dB]

-2

10

DFDMA

10

12

LFDMA

-3

10

-4

8

IFDMA

10

0

Dotted lines: no PS Dashed lines: RRC PS Solid lines: RC PS 2

0

4

6 PAPR [dB]

8

10

12

0

(a)

(b)

Figure 6.2: Comparison of CCDF of PAPR for IFDMA, DFDMA, LFDMA, and OFDMA with total number of subcarriers M = 512, number of input symbols N = 128, IFDMA spreading factor Q = 4, DFDMA spreading factor Qɶ = 2, and α (roll-off factor) = 0.22; (a) QPSK; (b) 16-QAM. PAPR. We denote 99.9-percentile PAPR as PAPR99.9%. Table 6.1 summarizes the 99.9-percentile PAPR for each subcarrier mapping. Table 6.1: 99.9-percentile PAPR for IFDMA, DFDMA, LFDMA, and OFDMA Modulation

Pulse shaping None

QPSK

16-QAM

IFDMA

DFDMA

LFDMA

OFDMA

0 dB

7.7 dB

7.7 dB

11.1 dB

RC

6.2 dB

7.7 dB

8.0 dB

N/A

RRC

5.3 dB

7.8 dB

8.7 dB

N/A

None

3.2 dB

8.7 dB

8.7 dB

11.1 dB

RC

7.8 dB

8.7 dB

9.0 dB

N/A

RRC

7.2 dB

8.7 dB

9.5 dB

N/A

* RC: raised-cosine pulse shaping, RRC: squared-root raised-cosine pulse shaping

79 CCDF of PAPR: QPSK, N = 256, N fft

0

occupied

= 64

CCDF of PAPR: 16-QAM, N = 256, N

10

10

IFDMA

occupied

-1

-1

0

0

Pr(PAPR>PAPR )

10

= 64

LFDMA

IFDMA

LFDMA

10 Pr(PAPR>PAPR )

fft

0

-2

10

α=1 α=0.8

-3

10

-4

10

0

α=0.2

α=0.6

α=0.4

α=0

10

α=0.8 10

Solid lines: without pulse shaping Dotted lines: with pulse shaping 2

4 6 PAPR [dB] 0

(a)

8

10

α=1

-2

10

α=0.6 α=0.4 α=0.2

-3

-4

0

Solid lines: without pulse shaping Dotted lines: with pulse shaping 2

α=0

4 6 PAPR [dB]

8

10

0

(b)

Figure 6.3: Comparison of CCDF of PAPR for IFDMA and LFDMA with M = 256, N = 64, Q = 4, and α (roll-off factor) of 0, 0.2, 0.4, 0.6, 0.8, and 1; (a) QPSK; (b) 16-QAM.

We can see that all the cases for SC-FDMA have indeed lower PAPR than that of OFDMA. Also, IFDMA has the lowest PAPR, and DFDMA and LFDMA have very similar levels of PAPR. Figure 6.3 shows the impact of the roll-off factor α on the PAPR when using raised-cosine pulse shaping for M = 256, N = 64, and Q = 4. We can see that this impact is more obvious in the case of IFDMA. As the roll-off factor decreases from 1 to 0, PAPR increases significantly for IFDMA. This observation is consistent with the results of the instantaneous peak power for IFDMA with pulse shaping in section 5.1 of chapter 5. This implies that there is a tradeoff between PAPR performance and out-of-band radiation since out-of-band radiation increases with increasing roll-off factor.

80

6.2.

PAPR of Multiple Antenna Transmission Signals

In this section, we analyze the PAPR characteristics of multiple antenna transmission for SCFDMA system. Specifically, we investigate unitary precoded transmit eigen-beamforming (TxBF) for 2 transmit and 2 receiver antennas which we described in section 4.3 of chapter 4. Since it is difficult to derive the PAPR analytically, we resort to numerical analysis using Monte Carlo simulation to investigate the PAPR properties of our MIMO SC-FDMA system. We use the parameters in section 4.3 of chapter 4 for the simulations in this section. As described in section 4.3 of chapter 4, we apply unitary precoding in the frequency domain after DFT. Precoding in the frequency domain is convolution (filtering) and summation in the time domain as shown in Figure 6.4. Thus we expect to see an increase in the peak power because of the filtering and summation.

 V11,k V12,k  S   , S k =  1,k  Vk =  V21,k V22,k   S 2,k  X k = Vk ⋅ S k  V11,k V12,k   S1,k  ⋅  =   S  V V 22, k   2, k   21,k  V11,k ⋅ S1,k + V12,k ⋅ S 2,k   =   V ⋅ S + V ⋅ S 22, k 2, k   21,k 1,k Frequency domain

Vij = {Vij ,0 ,Vij ,1 ,⋯,Vij , M −1} Sl = {Sl ,0 , Sl ,1 ,⋯, Sl , M −1} vij = IDFT {Vij } , sl = IDFT {Sl }  v11 * s1 + v12 * s2  x=   v21 * s1 + v22 * s2    Time domain

Figure 6.4: Precoding in the frequency domain is convolution and summation in the time domain.k refers to the subcarrier number.

81 10

-1

1st antenna (16-QAM)

Precoded

2nd antenna (QPSK)

0

Pr(PAPR>PAPR )

10

CCDF of PAPR for Unitary Precoded TxBF

0

10

-2

No precoding

Only summation 10

-3

2nd stream zero

10

-4

4

6

8 PAPR [dB]

10

12

0

Figure 6.5: CCDF of PAPR for 2x2 unitary precoded TxBF. Figure 6.5 shows the CCDF of the PAPR for 2x2 unitary precoded TxBF. To verify that the filtering aspect of the precoding is the dominant factor in increasing the PAPR, we also show results for the case when there is only filtering by intentionally setting the 2nd antenna signal to zero, that is, s2 = 0 (the graph labeled “2nd stream zero”) and for the case when there is only summation by intentionally replacing the convolution with a scalar multiplication of the first element of vij (the graph labeled “Only summation”). With precoding, we can see that there is an increase of 1.6 dB for the 1st antenna and 2.6 dB for the 2nd antenna in the 99.9-percentile PAPR. We can also observe that the filtering aspect is the main contributor to the increase of PAPR for precoding.

82 10

10

10

10

10

TxBF (no avr. & no quant.)

No precoding (QPSK)

-1

0

Pr(PAPR>PAPR )

10

0

-2

No precoding (16-QAM)

-3

TxBF (both avr. & quant.)

-4

TxBF (only quant.) TxBF ( only avr.)

-5

4

6

8 PAPR [dB]

10

12

0

Figure 6.6: Impact of quantization and averaging of the precoding matrix on PAPR.

10

-1

TxBF (no avr. & no quant.)

SM

0

Pr(PAPR>PAPR )

10

0

10

10

10

-2

SFBC (QPSK) SFBC (16-QAM) -3

-4

4

TxBF (avr. & quant.)

6

8 PAPR [dB] 0

Figure 6.7: PAPR comparison with other MIMO schemes.

10

12

83

Figure 6.6 illustrates the impact of quantization and averaging of the precoding matrix on the PAPR characteristics. We use direct quantization and averaging process described in section 4.3.1 of chapter 4. We average the channel H over 25 continuous subcarriers and perform direct quantization of the precoder matrix V using 3 bits (1 bit for amplitude and 2 bits for phase information). Without averaging of channel and quantization of precoding matrix, the MIMO TxBF signal has 1.6 ~2.6 dB higher PAPR99.9% with respect to single antenna transmission. When we apply both averaging and quantization, PAPR99.9% decreases by 0.7 dB. This is due to the fact that the averaging and quantization of the precoding matrix smoothes the time filtering, thus reduces the peak power. Figure 6.7 compares the CCDF of the PAPR for unitary precoded TxBF with those of different MIMO schemes. We consider space-frequency block coding (SFBC) and non-precoded spatial multiplexing (SM) for comparison. Since the spatial multiplexing that we consider does not have any precoding or spatial processing at the transmitter, it has the same PAPR as the case of single antenna transmission. Without quantization/averaging of precoding matrix, TxBF has higher PAPR99.9% than that of SFBC by 0.5 ~ 1 dB. However, TxBF and SFBC have similar PAPR when we have quantization/averaging of the precoding matrix.

6.3.

Peak Power Reduction by Symbol Amplitude Clipping

One way to reduce PAPR is to limit or clip the peak power of the transmitted symbols [55]. Depending on the smoothness of the limiter, we can define three types of limiter; hard, soft, and smooth limiter [56]. The input-out relationship of each type of amplitude limiter for a complex

84

symbol is as follows. Note that we leave the phase of the symbol as is and only modify the amplitude part. •

Hard limiter:

ym = g hard ( xm ) = Amax e j∡xm •

(6.5)

Soft limiter:

 xm , xm ≤ Amax ym = g soft ( xm ) =  j ∡xm  Amax e , xm > Amax •

(6.6)

Smooth limiter:

ym = g smooth ( xm ) = Amax erf ( xm / Amax ) e j∡xm

(6.7)

where xm is the input symbol to the limiter, ∡xm is the phase component of xm, ym is the output symbol of the limiter, Amax > 0, and erf ( x ) =

2

π



x

0

e − t dt . Figure 6.8 graphically 2

illustrates the input amplitude-output amplitude relationship of each limiter. In the subsequent analysis, we use the soft limiter for clipping. The problems associated with clipping are in-band signal distortion and generation of outof-band signal. Because SC-FDMA modulation spreads the information data across all the modulated symbols, in-band signal distortion is mitigated when an SC-FDMA symbol is clipped. To analyze the impact of symbol amplitude clipping on the link level performance, we apply clipping to the unitary precoded TxBF MIMO system that we described in chapter 4 and show numerical simulation results. We use the parameters and assumptions in section 4.3 of chapter 4

85

ym Soft limiter Hard limiter

Amax

Smooth limiter

Amax

xm

Figure 6.8: Three types of amplitude limiter. for the simulations in this section. Figure 6.9 shows the block diagram of a symbol amplitude clipping method for spatial multiplexing SC-FDMA MIMO transmission for Mt transmit antenna. In our current analysis, we apply the clipping after baseband pulse shaping. We can also consider other sophisticated clipping methods, such as iterative clipping. Figure 6.10 shows the CCDF of symbol power with clipping at various levels. With 7 dBmax clipping less than 1% of the symbols are clipped. Note that even with as much as 3 dB-max PAPR clipping, only about 10% of the modulated symbols are clipped. Figure 6.11 shows the uncoded bit error rate (BER) and coded frame error rate (FER) performances when we apply symbol amplitude clipping. We can observe that the performance degradation due to clipping is almost none for 7 dB-max clipping. The performance degrades

86

{x0,n ,⋯, x0, N −1}

Sucarrier Mapping

Add CP/ PS

IDFT

Spatial Processing

DFT

{xM t −1,n ,⋯, xM t −1, N −1}

Amplitude Clipping

DAC/ RF

Channel

Sucarrier Mapping

DFT

Add CP/ PS

IDFT

Amplitude Clipping

DAC/ RF

Figure 6.9: Block diagram of a symbol amplitude clipping method for SC-FDMA MIMO transmission with Mt transmit antenna.

10

10

-1

No clipping

-2

m

Pr(|y |2>w)

10

CCDF of Symbol Power

0

10

10

-3

3 dB-max clipping 5 dB-max clipping

-4

7 dB-max clipping 0

2

4

6

8

10

w [dB]

Figure 6.10: CCDF of symbol power after clipping. ym represents the baseband symbol. Solid line represents CCDF for antenna 1 and dashed line represents CCDF for antenna 2.

87 0

0

10

10

-1

10 -2

FER

Raw BER

10

-2

10

-4

10

No clipping 7 dB-max clipping 5 dB-max clipping 3 dB-max clipping

-3

10

-4

-6

10

0

No clipping 7 dB-max clipping 5 dB-max clipping 3 dB-max clipping

5

10 15 20 SNR per Antenna [dB]

25

30

10

0

2

4 6 SNR per Antenna [dB]

(a)

(b)

Figure 6.11: Link level performance for clipping; (a) uncoded BER; (b) coded FER.

0 3 dB-max clipping

-10

Power [dB]

-20 -30

5 dB-max clipping 7 dB-max clipping

-40 -50 No clipping -60 -4

-3

-2 -1 0 1 2 Normalized Frequency Band

Figure 6.12: PSD of the clipped signals.

3

4

8

10

88

slightly more when we apply 5 or 3 dB-max clipping. For 3 dB-max clipping, an error floor starts to appear in high SNR region for uncoded BER. But when we incorporate forward error correction coding, it mitigates this effect. Thus, we can use a modest amount of amplitude clipping to reduce the PAPR for unitary precoded TxBF with SC-FDMA without compromising the link level performance. Clipping, consequently, will generate both in-band and out-of-band frequency components. Figure 6.12 shows the power spectral density (PSD) of the clipped signals. For PSD calculation, we use Hanning window with 1/4 of window overlapping. For 7 dB-max clipping, the spectrum is almost the same as that of the original signal. More pronounced out-of-band components arise when we use 5 or 3 dB-max clipping. Thus, we should control the amount of clipping depending on the out-of-band radiation requirements.

6.4.

Summary and Conclusions

In this chapter, we analyzed the PAPR of SC-FDMA signals for single and multiple antenna transmissions using numerical simulations. We showed that SC-FDMA signals indeed have lower PAPR compared to OFDMA signals. Also, we have shown that DFDMA and LFDMA incur higher PAPR compared to IFDMA but compared to OFDMA, it is still lower, though not significantly. Another noticeable fact is that pulse shaping increases PAPR, thus degrades the power efficiency and that the roll-off factor in the case of raised-cosine pulse shaping has a significant impact on PAPR of IFDMA. A pulse shaping filter should be designed carefully in order to reduce the PAPR without degrading the system performance. For multiple antenna transmission, we observed that unitary precoding TxBF scheme

89

increases the PAPR because of its filtering aspect. It was interesting to find out that averaging and quantization of the precoder matrix actually reduces the PAPR. In order to reduce the peak power in a SC-FDMA system, we proposed a symbol amplitude clipping method and applied it to the 2x2 unitary precoded TxBF system. Numerical simulation results showed that moderate clipping does not affect the link level performance. But out-ofband radiation is a concern when we use clipping.

Chapter 7

ChannelChannel-Dependent Scheduling of Uplink SCSC-FDMA Systems

As we illustrated in section 2.1 of chapter 2, wide band wireless channels experience time and frequency-selective fading because of user’s mobility and multi-path propagation. In wide band multi-user uplink communications, the channel gain of each user is different for different time and frequency subcarrier when the channels are uncorrelated among users. Time and frequency resources which are in deep fading for one user may be in excellent conditions for other users. The resource scheduler in the base station can assign the timefrequency resources to a favorable user which will increase the total system throughput [57], [58], [59]. We term the class of this adaptive resource scheduling method as channeldependent scheduling (CDS) which can greatly increase the spectral efficiency (bits/Hz). Another way to exploit the time and frequency-selectivity in the channel to its advantage is the adaptive modulation and coding (AMC) scheme along with subband or multicarrier communications. AMC dynamically adapts the modulation constellation and the channel coding rate depending on the channel condition and thus improves the energy efficiency and the data rate [60], [61]. By applying AMC on subbands or subcarriers, we can dramatically

90

91

improve the link quality and increase the data rate for time and frequency-selective fading channel. In this chapter, we first give a general overview of CDS in an uplink SC-FDMA system. We also analyze the capacity for the distributed (IFDMA) and localized subcarrier mapping schemes. In section 7.2, we investigate the impact of imperfect channel state information (CSI) on CDS. We analyze the data throughput of an SC-FDMA system with uncoded adaptive modulation and CDS when there is a feedback delay. In section 7.3, we propose a hybrid subcarrier mapping scheme using direct sequence spreading technique on top of SCFDMA modulation. We show the throughput improvement of the hybrid subcarrier mapping over the conventional subcarrier mapping schemes.

7.1.

Channel-Dependent Scheduling in an Uplink SC-FDMA System

In this section, we investigate channel-dependent resource scheduling for an SC-FDMA system in uplink communications. Specifically, we analyze the capacity gains from CDS for the two different flavors of subcarrier mapping of SC-FDMA. For distributed subcarrier mapping, we consider IFDMA. A key question of CDS is how we should allocate time and frequency resources fairly among users while achieving multi-user diversity and frequency selective diversity. To do so, we introduce utility-based scheduling where utility is an economic concept representing level of satisfaction [62], [63], [64], [65], [66]. The choice of a utility measure influences the tradeoff between overall efficiency and fairness among users. In our studies, we consider two different utility functions: aggregate user throughput for maximizing system capacity and aggregate logarithmic user throughput for maximizing proportional fairness. The

92

objective is to find an optimum resource assignment for all users in order to maximize the sum of user utility at each transmission time interval (TTI). If the user throughput is regarded as the utility function, the resource allocation maximizes rate-sum capacity ignoring fairness among users. Therefore, only the users near the base station who have the best channel conditions occupy most of the resources. On the other hand, setting the logarithmic user data rate as the utility function provides proportional fairness. In considering the optimization problem of CDS for multicarrier multiple access, it is theoretically possible to assume that the scheduler can assign subcarriers individually. Allocating individual subcarriers is, however, a prohibitively complex combinatorial optimization problem in systems with a large number of subcarriers and user terminals. Moreover, assigning subcarriers individually would introduce unacceptable control signaling overhead. In practice, we allocate the resource in chunks, which are disjoint sets of subcarriers. We refer to this chunk as a resource unit (RU). As a practical matter for SC-FDMA, chunkbased transmission is desirable since the input data symbols are grouped into a block for DFT operation before subcarrier mapping. We will consider only chunk-based scheduling in the remainder of this section. With regards to chunk structure, consecutive subcarriers comprise a chunk for LFDMA and distributed subcarriers with equidistance for IFDMA. Even with subcarriers assigned in chunks, optimum scheduling is extremely complex for two reasons: 1) The objective function is complicated, consisting of nonlinear and discrete constraints dependent on the combined channel gains of the assigned subcarriers; and 2) there is a total transmit power constraint for each user. Furthermore, the optimum solution entails

93

combinatorial comparisons with high complexity. Instead of directly solving the optimization problem, we can use a sub-optimal chunk allocation scheme for both IFDMA and LFDMA to obtain most of the benefits of CDS. For LFDMA, we can apply a greedy chunk selection method where each chunk is assigned to the user who can maximize the marginal utility when occupying the specific chunk. For IFDMA, we can achieve the benefit of multi-user diversity by selecting users in order of the estimated marginal utility based on the average channel condition over the entire set of subcarriers. The users with higher channel gains may occupy a larger number of chunks than users with lower channel gains. The details of the chunk selection method are in [64] and [65]. Our throughput measure in this study is the sum of the upper bound on user throughputs given by Shannon’s formula, C = BW⋅log(1+SNR) where BW is the effective bandwidth depending on the number of occupied subcarriers and SNR is signal-to-noise ratio of a block. Figures 7.1 and 7.2 are the results of computer simulations of SC-FDMA with 256 subcarriers spread over a 5 MHz band. They compare the effects of channel dependent subcarrier allocation (S) with static (round-robin) scheduling (R) for LFDMA and IFDMA. In all of the examples, the scheduling took place with chunks containing 8 subcarriers and we assume perfect knowledge of the channel state information. Figure 7.1 shows two effects of applying different utility functions in the scheduling algorithm. In the two graphs in Figure 7.1, the utility functions are the sum of user throughputs and the sum of the logarithm of user throughputs, respectively. Each graph

94 45

45

35

40 Aggregate throughput [Mbps]

Aggregate throughput [Mbps]

40

R-LFDMA S-LFDMA R-IFDMA S-IFDMA

30 25 20 15

30 25 20 15 10

10 5 4 8 16

35

R-LFDMA S-LFDMA R-IFDMA S-IFDMA

32

64 Number of users

(a)

128

5 4 8 16

32

64 Number of users

128

(b)

Figure 7.1: Comparison of aggregate throughput with M = 256 system subcarriers, N = 8 subcarriers per user, bandwidth = 5 MHz, and noise power per Hz = -160 dBm; (a) utility function: sum of user throughputs; (b) utility function: sum of the logarithm of user throughputs. shows the aggregate throughput as a function of the number of simultaneous transmissions. Figure 7.2 shows the expected user throughput at each distance from the base station for the same utility functions. The simulation results in the figures use the following abbreviations: RLFDMA (static round robin scheduling of LFDMA), S-LFDMA (CDS of LFDMA), RIFDMA (static round robin scheduling of IFDMA), and S-IFDMA (CDS of IFDMA). Figure 7.1 shows that for throughput maximization (utility = bit rate), the advantage of channel dependent scheduling over round robin scheduling increases as the number of users increases. This is because the scheduler selects the closer users who can transmit at higher data rate. If there are more users, the possibility of locating users at closer distance to the base station increases. As a result, the CDS achieves significant improvements for both IFDMA

95 5 R-LFDMA S-LFDMA R-IFDMA S-IFDMA

4

Average user data rate [Mbps]

Average user data rate [Mbps]

5

3

2

1

0 0.2

0.4 0.6 0.8 User distance from target BS [km]

(a)

1

R-LFDMA S-LFDMA R-IFDMA S-IFDMA

4

3

2

1

0 0.2

0.4 0.6 0.8 User distance from target BS [Km]

1

(b)

Figure 7.2: Average user data rate as a function of user distance with M = 256 system subcarriers, N = 8 subcarriers per user, bandwidth = 5 MHz, and noise power per Hz = -160 dBm; (a) utility function: sum of user throughputs; (b) utility function: sum of the logarithm of user throughputs. and LFDMA. In the case of logarithmic rate utility, the CDS gain stops increasing beyond approximately 32 users. With 32 users, maximizing logarithmic rate utility can increase system capacity by a factor of 1.8 for LFDMA and 1.26 for IFDMA relative to static scheduling. Figure 7.2 shows that the CDS scheme based on the logarithm of user throughput as a utility function provides proportional fairness whose gains are shared among all users, whereas the CDS gains are concentrated to the users near the base station when the user throughput is considered as the utility function. For static subcarrier assignment (round robin scheduling), a system with users each transmitting at a moderate data rate is better off with IFDMA while LFDMA works better in a system with a few high date rate users. Due to the advantages of lower outage probability

96

and lower PAPR, IFDMA is an attractive approach to static subcarrier scheduling. For CDS, LFDMA has the potential to achieve considerably higher data rate. The results show that CDS increases system throughput by up to 80% relative to static scheduling for LFDMA but the increase is only 26% for IFDMA. We can exploit the scheduling gains in LFDMA to reduce power consumption and PAPR by using power control to establish a power margin instead of increasing system capacity.

7.2.

Impact of Imperfect Channel State Information on CDS

In section 7.1, we assumed perfect knowledge of the channel state information (CSI). In a practical wireless system, CSI is often not known at all or only part of the information is available due to limited feedback mechanism and channel estimation error. Also, the CSI may become outdated because of the feedback delay. In a centralized resource management scheme, the base station makes all the decision regarding the uplink transmission parameters based on the uplink channel quality and feeds back the transmission parameter information to the mobile terminal. A delay in the feedback mechanism is inevitable in a practical feedback system and the feedback delay has a major impact on the system performance especially when the channel is fading rapidly. In this section, we investigate the impact of imperfect CSI on CDS in uplink communications by considering outdated CSI due to feedback delay. We assume the channel estimation is perfect and also assume there is no error during the feedback signaling. Since there is no general closed form for the capacity equation for outdated CSI, we resort to numerical simulations in the context of an SC-FDMA system with uncoded adaptive

97 Mobile terminals

Base station

User K

Channel K User 2

DFT

Subcarrier Mapping

Constellat ion Mapping

User 1 Channel 2 IDFT

CP / PS

Channel 1

SC-FDMA Receiver

Resource Scheduler

Data flow Control signal flow

Figure 7.3: Block diagram of an uplink SC-FDMA system with adaptive modulation and CDS for K users.

modulation to measure the data throughput. Figure 7.3 describes the system under investigation. We consider K number of users orthogonally and independently sending data to the base station and we assume that the channel responses are independent of each other. The resource scheduler in the base station performs the following two tasks upon acquiring the channel information for all users. First, the scheduler searches for an optimal set of users that maximizes the total capacity. Next, for each user, it decides the modulation constellation based on the SNR of the chunk allocated to the user. The scheduler, then, sends the chunk allocation and modulation information back to

98

each user via the feedback control signaling channel. When there is a delay in the feedback channel, it creates two sources of performance degradation. First, the chunk allocation becomes outdated by the time the users transmit and it no longer is the optimal allocation. Second, the modulation constellation for each user no longer matches the channel condition at the time of transmission. Because of these two factors, we can expect significant decrease in the total capacity for time-varying channels. In our analysis, we introduce a time delay between the time instance of the channel estimation (t1) and the actual time of data transmission from the mobile terminals (t2). We consider a quasi-static multipath fading channel in which channel response is constant over a TTI but varies between each TTI depending on a correlation parameter. For a Rayleigh fading channel, we represent the correlation coefficient for time delay ∆t = t2 - t1 as

ρ (∆t ) = J 0 (2π v∆t / λ ) = J 0 (2π f D ∆t )

(7.1)

where J 0 (i) is the zero-order Bessel function, fD = v/λ is the maximum Doppler frequency, v is the velocity of the mobile terminal, and λ is the wavelength of the carrier signal [11]. We model the multipath fading channel at t1 with an R-path discrete time channel impulse response, h (t1 ) ( n) , as follows. R −1  τ  h (t1 ) ( n) = ∑ A ⋅ Prel (τ r ) ⋅ wr(t1 ) ⋅ δ  n − r  Ts  r =0 

(7.2)

where { wr(t1 ) }’s are zero mean complex Gaussian i.i.d. noise process, τr is the propagation

99

delay of path r, Ts is the symbol duration, A is a normalization parameter such that the average power of the channel equals to 1, Prel(τr) is the relative power of the delay profile model for path r, and δ(n) is the discrete time Dirac-delta function. Note that {τr}’s are integer-multiples of Ts. Using (7.1), we can generate the multipath fading channel at t2 , h (t2 ) (n) , as follows. R −1

τr

r =0

Ts

h (t2 ) ( n) = ∑ A ⋅ Prel (τ r ) ⋅ wr(t2 ) ⋅ δ (n −

)

(7.3)

and

wr(t2 ) = ρ ( ∆t ) ⋅ wr(t1 ) + 1 − ρ 2 ( ∆t ) ⋅ nr

(7.4)

where nr is a zero mean complex Gaussian noise process. We will use (7.2) and (7.3) to analyze the effect of the outdated channels on our system. In [64], we derived the SNR of user k, γk, as follows based on [67] for an MMSE equalizer when a set of subcarriers Isub,k is assigned to the user.   γk =   1  N

  1 − 1 γ i ,k  ∑  i∈I sub ,k γ i , k + 1 

−1

(7.5)

where N is the number of subcarriers allocated to the user and γ i ,k is the SNR of subcarrier i for user k. For equal power allocation scheme, we can express γ i ,k as

100

γ i ,k =

( Pk / N ) ⋅ H i ,k σ i2

(7.6)

where Pk is the total transmit power of user k, Hi,k is the channel gain of subcarrier i for user k, and σ i2 is the noise power of subcarrier i.

In our simulations, we apply the effect of the time delay during feedback in the following manner. We first calculate the received SNRs for the subcarrier allocations and select the modulation constellations based on the channels at time t1 and then calculate the received SNRs for the throughput calculations based on the channels at time t2. For the numerical simulations, we consider the following setup and assumptions. •

Carrier frequency: 2 GHz.



Transmission bandwidth: 5 MHz.



Duration of a TTI: 0.5 ms.



Blocks per TTI: 7. We assume there is no CP in a block for simplicity of the simulation.



The size of the subcarrier chunk is the same among all users and each user occupies only one subcarrier chunk.



The number of users is K, the total number of subcarriers M, the size of a chunk N, the number of chunks Q, and M = Q⋅N.



Channel: 3GPP TU6 i.i.d. quasi-static Rayleigh fading channel [12]. We assume that the channel is constant during the duration of a TTI. We also assume that all the users have the same path loss to the base station.

101



Modulation format: Quadrature amplitude modulation (QAM). We consider SQAM where S = 2B and B is the number of bits per symbol. We have 8 classes of QAM where B ∈ {1, 2, … , 8}. B = 1 corresponds to BPSK and B = 2 to QPSK.



We assume equal transmit power and equal receive noise power for all users.



We use aggregate user throughput as the utility function of CDS.

We define the system throughput as the number of information bits per second received without error. Let user k be assigned Sk-QAM where S k = 2 Bk and Bk is the number of bits per symbol. Then we express the system throughput for user k as follows [68], [69]. TPk =

7 ⋅ N ⋅ Bk ⋅ PSR ( S k , γ k ) TTTI

(7.7)

where TTTI is the duration of a TTI, γk is the received SNR for user k, and PSR ( i ) is the packet success rate (PSR) which we define as the probability of receiving an information packet correctly. The aggregate throughput TPtotal is the sum of the K individual throughput measures in (7.7). For an uncoded packet with a size of L bits, PSR becomes PSR ( S , γ ) = {1 − Pe ( S , γ )}

L

(7.8)

where Pe ( S , γ ) is the BER for a given constellation size S and SNR γ. For an additive white Gaussian noise (AWGN) channel with S-QAM modulation and ideal coherent detection, we can upper-bound Pe ( S , γ ) as follows [70].

102

Pe ( S , γ ) ≤ 0.2e

−1.5γ ( S −1)

(7.9)

Using this upper bound, PSR in (7.8) becomes −1.5γ   ( S −1)  PSR ( S , γ ) = 1 − 0.2e   

L

(7.10)

and the system throughput for user k becomes 7 ⋅ N ⋅ Bk TPk = TTTI

−1.5γ k   ⋅ 1 − 0.2e ( Sk −1)   

L

(7.11)

Then, the total system throughput for K number of users is K

TPtotal = ∑ TPk

(7.12)

k =1

Figure 7.4 shows the system throughput vs. SNR for the 8 classes of QAM. We use L = 100 bits per packet and N = 32 subcarriers per chunk. Based on Figure 7.4, we set the SNR boundaries for the adaptive modulation. Table 7.1 describes the SNR boundaries for our analysis.

103 4 B=8

3.5

B=7

Throughput [Mbps]

3 B=6

2.5

B=5 2 B=4 1.5 B=3 1

B=2

0.5 0

B=1

0

5

10

15

20 25 SNR [dB]

30

35

40

Figure 7.4: System throughput vs. SNR for the 8 classes of QAM. B is the number of bits per symbol.

Table 7.1: SNR boundaries for adaptive modulation. SNR (dB)

Bits per symbol (B)

Modulation

> 8.7

1

BPSK

8.7 ~ 13.0

2

QPSK

13.0 ~ 16.7

3

8-QAM

16.7 ~ 20.0

4

16-QAM

20.0 ~ 23.3

5

32-QAM

23.3 ~ 26.6

6

64-QAM

26.6 ~ 29.6

7

128-QAM

29.6 <

8

256-QAM

104 mobile speed = 3 km/h (f = 5.6 Hz)

mobile speed = 60 km/h (f = 111 Hz)

D

D

18

14 12

Aggregate throughput [Mbps]

LFDMA: Static LFDMA: CDS IFDMA: Static IFDMA: CDS

16

10 8 6 4 2 0

1

2 3 Feedback delay [ms]

4

5

LFDMA: Static LFDMA: CDS IFDMA: Static IFDMA: CDS

16 14 12 10 8 6 4 2 0

1

2 3 Feedback delay [ms]

(a)

4

5

(b)

Figure 7.5: Aggregate throughput with CDS and adaptive modulation; (a) mobile speed = 3 km/h (Doppler = 5.6 Hz); (b) mobile speed = 60 km/h (Doppler = 111 Hz). mobile speed = 60 km/h (f = 111 Hz) D

18 LFDMA: Static LFDMA: CDS IFDMA: Static IFDMA: CDS

16 Aggregate throughput [Mbps]

Aggregate throughput [Mbps]

18

14 12 10 8 6 4 2 0 0

1

2 3 Feedback delay [ms]

4

5

Figure 7.6: Aggregate throughput with CDS and constant modulation (16-QAM) with mobile speed of 60 km/h.

105 Feedback delay = 3 ms 18 LFDMA: Static LFDMA: CDS IFDMA: Static IFDMA: CDS

Aggregate throughput [Mbps]

16 14 12 10 8 6 4 2 0

20 (37) 40 (74) 60 (111) Mobile speed [km/h] (Doppler [Hz])

80 (148)

Figure 7.7: Aggregate throughput with CDS and adaptive modulation with feedback delay of 3 ms and different mobile speeds. Figures 7.5 and 7.6 show the results of the Monte Carlo simulations of the investigated system. Total number of subcarriers is M = 256, the number of chunks is Q = 16, the number of subcarriers per chunk is N = 16, and total number of users is K = 64. We perform 2000 independent iterations of the simulation. Figure 7.5 shows the aggregate throughputs with CDS and adaptive modulation for different mobile speeds. For low mobile speed, we can see that the feedback delay does not affect the system throughput. However, at high mobile speed, the throughput degrades as the feedback delay increases and the degradation is most significant for LFDMA with CDS. This implies that LFDMA is very sensitive to the quality of CSI when we apply CDS for fast fading

106

channel. Another interesting observation is that the static round-robin scheduling also suffers from throughput decrease. This is due to the fact that even though there is no chunk allocation mismatch, outdated CSI causes the adaptive modulation to be mismatched with the current channel condition. Figure 7.6 shows the aggregate throughput with CDS for mobile speed of 60 km/h and constant modulation (16-QAM) instead of adaptive modulation. It further verifies the observation in Figure 7.5 (b) for static scheduling that the cause of the throughput decrease is the adaptive modulation process. We do not see any throughput decrease for the static scheduling when we use constant modulation in the same channel condition. Figure 7.7 shows the aggregate throughput with CDS and adaptive modulation with feedback delay of 3 ms and different mobile speeds. We can see the performance impact of the mobility of the user on each type of scheduling method. Similarly with Figure 7.5 (b), LFDMA with CDS suffers the most throughput decrease when the mobile speed is high.

7.3.

Hybrid Subcarrier Mapping

As we can see from Figure 7.7, localized subcarrier mapping yields much higher throughput for low speed users when we use CDS but the throughput of LFDMA with CDS becomes lower when the users are moving at high speed. In this situation, static round-robin scheduling using distributed subcarrier mapping is advantageous since it requires no overhead and no computation. Also, IFDMA has an additional advantage in PAPR. Thus, localized subcarrier mapping with CDS is more advantageous for low mobility users and IFDMA with static scheduling is better for users with high mobility. When there are low mobility users and high

107

mobility users at the same, accommodating both types of subcarrier mapping can lead to higher data capacity. Conventional SC-FDMA cannot efficiently accommodate both distributed and localized mapping schemes since subcarriers must not overlap to maintain orthogonality among users. Using orthogonal direct sequence spread spectrum technique prior to SC-FDMA modulation, both mapping can coexist with overlapping subcarriers as illustrated in Figure 7.8.

Terminal 1 Terminal 2 Terminal 3

Spreading Code 1

subcarriers

Spreading Code 2

Conventional

subcarriers Hybrid

Figure 7.8: Conventional subcarrier mapping and hybrid subcarrier mapping. We refer to this multiple access scheme as single carrier code-frequency division multiple access (SC-CFDMA). Figure 7.9 shows the block diagram of an SC-CFDMA system and Figure 7.10 compares SC-FDMA and SC-CFDMA in terms of occupied subcarriers for the same number of users.

108

Npoint DFT

Spreading

Subcarrier Mapping

Mpoint IDFT

SC-FDMA Modulation

Add CP/ PS

Channel

Remove CP

SC-FDMA Demodulation

Mpoint DFT

Subcarrier Demapping/ Equalization

Despreading

Detect

Npoint IDFT

Figure 7.9: Block diagram of an SC-CFDMA system.

User 1

User 1 Code 1

User 2 User 3 User 4

Code 2 subcarriers SC-FDMA Q (# of users) = 4

User 2 User 3 User 4

subcarriers SC-CFDMA Q (# of users) = Qcode×Qfrequency= 2×2 = 4

Figure 7.10: Comparison between SC-FDMA and SC-CFDMA in terms of occupied subcarriers for the same number of users.

109 Aggregate throughput [Mbps]

16 IFDMA: static Hybrid LFDMA: CDS

14 12 10 8 6 4 2 0 25/75

50/50

75/25

Low/high mobility users percentile [%/%]

Figure 7.11: Aggregate throughputs for hybrid subcarrier mapping method and other conventional subcarrier mapping methods with CDS and adaptive modulation. Figure 7.11 compares the aggregate throughput between hybrid subcarrier mapping method and conventional subcarrier mapping methods when we apply CDS and adaptive modulation. In hybrid subcarrier mapping method, we apply LFDMA with CDS to low mobility users and IFDMA with static scheduling to high mobility users. Total number of subcarriers is M = 256, the number of chunks is Q = 16, the number of subcarriers per chunk is N = 16, the total number of users is K = 64, and the feedback delay is 3 ms. Low mobility users have velocity of 3 km/h and high mobility users have velocity of 60 km/h. In the figure, the label ‘X/Y’ means that X-percentile of the users have low mobility and Y-percentile of the users have high mobility. Clearly, when high mobility users are the majority of the entire users (the graphs labeled ‘25/75’ and ‘50/50’), hybrid subcarrier mapping scheme yields higher data throughput than the conventional subcarrier mapping schemes.

110

7.4.

Summary and Conclusions

In this chapter, we investigated channel-dependent scheduling (CDS) for an uplink SC-FDMA system. Specifically, we analyzed the capacity gains from CDS for the two different flavors of subcarrier mapping of SC-FDMA. We also looked into the impact of imperfect channel state information (CSI) on CDS by considering the feedback delay. We showed that localized subcarrier mapping yields highest aggregate data throughput when we use CDS. However, we also showed that LFDMA is very sensitive to the quality of CSI and the capacity gain quickly decreases when the channel changes very fast. For high mobility users, IFDMA with static round-robin scheduling is more suitable. To accommodate both low and high mobility users simultaneously, we proposed a hybrid subcarrier mapping method using orthogonal code spreading on top of SC-FDMA. We showed that the hybrid subcarrier mapping scheme has capacity gains over conventional subcarrier mapping schemes when there are both low and high mobility users at the same time.

Chapter 8

Conclusions

Single carrier FDMA (SC-FDMA) is a new multiple access scheme that is currently being adopted in uplink of 3GPP LTE. SC-FDMA utilizes single carrier modulation and frequency domain equalization, and has similar performance and essentially the same overall complexity as those of OFDMA system. A salient advantage over OFDMA is that the SC-FDMA signal has lower PAPR because of its inherent single carrier transmission structure. SC-FDMA has drawn great attention as an attractive alternative to OFDMA, especially in the uplink communications where lower PAPR greatly benefits the mobile terminal in terms of transmit power efficiency and manufacturing cost. In this thesis, we gave a detailed overview of an SC-FDMA system and compared it with the OFDMA system and also with the direct sequence CDMA (DS-CDMA) with frequency domain equalization (FDE) system. We also explained the different subcarrier mapping schemes in SC-FDMA; distributed and localized. The two flavors of subcarrier mapping schemes give the system designer a flexibility to adapt to the specific needs and requirements. Along with the introduction of SC-FDMA, we investigated the followings. •

Realization of MIMO spatial multiplexing in SC-FDMA with practical limitations: We

111

112

specifically analyzed the unitary precoded transmit eigen-beamforming (TxBF) technique on an SC-FDMA system. We showed with numerical simulations that there is a trade-off between the feedback overhead and the link performance loss. We also showed that feedback delay is another impairment that degrades the performance especially for the high speed mobile users. •

Analysis of the peak power characteristics of SC-FDMA signals: By using Chernoff bound and using Monte Carlo simulations, we characterized the peak power of an SC-FDMA signal both analytically and numerically. We saw that the peak power of an SC-FDMA signal for any subcarrier mapping is indeed lower than that of OFDM and observed that IFDMA has the lowest peak power among the considered subcarrier mapping schemes of SC-FDMA. For IFDMA, the roll-off factor of the raised-cosine pulse shaping filter has a great impact on the distribution of the instantaneous power. For LFDMA, the input block size is the major factor characterizing the distribution of peak power. For multiple antenna transmission, we observed that unitary precoded TxBF scheme increases the peak power because of its filtering aspect. Interestingly, averaging and quantization of the precoder matrix actually reduces the peak power.



Peak power reduction by symbol amplitude clipping: In order to reduce the peak power in an SC-FDMA system, we proposed a symbol amplitude clipping method and applied it to the 2x2 unitary precoded TxBF system. Numerical simulation results showed that moderate clipping does not affect the link level performance. But out-of-band radiation is a concern when we use clipping.

113



Channel-dependent resource scheduling for SC-FDMA: We investigated channel-dependent scheduling (CDS) for an uplink SC-FDMA system. Specifically, we analyzed the capacity gains from CDS for the two different flavors of subcarrier mapping of SC-FDMA. We also looked into the impact of imperfect channel state information (CSI) on CDS by considering the feedback delay. We showed that localized subcarrier mapping yields highest aggregate data throughput when we use CDS. However, we also showed that LFDMA is very sensitive to the quality of CSI and the capacity gain quickly decreases when the channel changes very fast. For high mobility users, IFDMA with static roundrobin scheduling is more suitable since it does not require any computation and the PAPR is low. To accommodate both low and high mobility users simultaneously, we proposed a hybrid subcarrier mapping method using orthogonal code spreading on top of SC-FDMA. We showed that the hybrid subcarrier mapping scheme has capacity gains over conventional subcarrier mapping schemes when there are both low and high mobility users at the same time.

For future work, we list the following items. •

Impact of carrier frequency offset on the system performance: In this thesis, we did not consider carrier frequency offset which is a common impairment at the receiver. Its impact on the link level performance and capacity is an important issue for a practical implementation.



Peak power characteristics of SC-FDMA pilot reference signals: We can apply the same numerical analysis to the pilot signals proposed by 3GPP LTE and characterize the peak

114

power characteristics. •

More sophisticated clipping methods, such as iterative clipping, for peak power reduction: We showed that clipping is a very effective way to reduce peak power without compromising the link performance. Yet, out-of-band radiation is a problem for clipping. In order to mitigate the out-of-band radiation associated with clipping, we can apply clipping before and after the pulse shaping and iterate the process until we reach a satisfactory level of performance.



Impact of channel estimation error on unitary precoded TxBF MIMO SC-FDMA system: In our analysis, we assumed that the channel estimation is without error. In real systems, channel estimation error is unavoidable and this will have a significant impact on our unitary precoded TxBF MIMO SC-FDMA system.



Impact of channel estimation error on CDS considering error correction coding: Channel estimation error adds to the performance degradation in adaptive modulation scheme and channel-dependent resource scheduling scheme. We can analyze its impact on capacity using our adaptive modulation/CDS system.

Appendix A

Derivations of the Upper Bounds Bounds in Chapter 5

A.1

Derivation of (5.8) and (5.9)

We can expand E {Zeν Z } as follows.

 ∞   ∞  ∞  ∞ E {Zeν Z } = E  ∑ ak exp ν ∑ al   = ∑ E ak ∏ eν al   l =−∞   k =−∞  l =−∞   k =−∞    ν ak ∞ ν al  = ∑ E  ak e ∏ e  k =−∞ l =−∞   l ≠k   ∞

=



∑ {

k =−∞

E ak eν ak



(A.1)

} ∏ E {e } ν al

l =−∞ l ≠k

We can also expand E {eν Z } similarly as follows.   ∞   ∞  E eν Z = E exp ν ∑ al   = E  ∏ eν al   l =−∞   l =−∞  

{ }



{ }

=∏E e l =−∞

ν al

By applying (A.1) and (A.2) to (5.7) for ν = νˆ , we get the following.

115

(A.2)

116 ∞

∑ {

k =−∞

E ak eνˆak







} ∏ E {e } − δ ⋅ ∏ E {e } = 0 νˆal

l =−∞ l≠k

νˆal

l =−∞





∑ E {a e } ∏ E {e } = δ ⋅ ∏ E {e } νˆak

k

k =−∞

νˆal

l =−∞ l ≠k

l =−∞



By dividing each side with

(A.3)

νˆal

∏ E {eν } , we obtain ˆal

l =−∞





k =−∞

{ } =δ E {e }

E ak eνˆak

(A.4)

νˆak

Using (A.2), (5.6) becomes Pr {Z ≥ δ } ≤ e

A.2

ˆ −νδ



∏ E {eν } ˆ ak

(A.5)

k =−∞

Derivations of (5.12) and (5.13)

From (5.11),

sk ∈ CBPSK

ak ∈ {− p (t0 − kT ), p (t0 − kT )}

(A.6)

Also, P {sk = −1} = P {sk = 1} =

1 2

1 P {ak = − p (t0 − kT )} = P {ak = p (t0 − kT )} = 2

Using (A.6) and (A.7), (5.8) becomes

(A.7)

117 K max



k =− K max

1 1 ˆ ˆ p (t0 − kT )eν p (t0 − kT ) − p (t0 − kT )e −ν p ( t0 − kT ) 2 2 =δ 1 νˆ p (t0 − kT ) 1 −νˆ p (t0 −kT ) e + e 2 2

(A.8)

After we determine νˆ from (A.8), the upper bound in (5.9) becomes ) Pub( BPSK ≜ 2e −νδˆ , SC

= 2e −νδˆ = 2e =e

ˆ −νδ

ˆ −νδ

∏ E {eν } K max

ˆak

k =− K max K max

 1 νˆ p (t0 − kT ) 1 −νˆ p (t0 − kT )  + e  e  2  k =− K max  2



1   2

1   2

2 K max +1

2 K max

∏ (e K max

νˆ p ( t0 − kT )

+e

−νˆ p ( t0 − kT )

k =− K max

∏ ( eν K max

ˆ p ( t0 − kT )

k =− K max

+ e−νˆ p (t0 − kT )

)

(A.9)

)

and we upper-bound the CCDF as 2 ) Pr  x ( BPSK ) (t0 , s ) ≥ w ≤ Pub( BPSK , SC  

A.3

(A.10)

Modified Upper Bound Derivation for QPSK

For complex modulation constellations, we can express (5.4) as follows from the probability regions in Figure A.1. 2 2 Pr  x(t0 , s ) ≥ w = Pr  Z ≥ w      w w 2 2   ≤ Pr  Z re ≥  + Pr  Z im ≥  2 2  

= Pr  Z re ≥ δ ′ + Pr  Z im ≥ δ ′

where

(A.11)

118

Im {Z }

w

Im {Z } = w

2

2

w − w

2

2

− w

Im {Z } = − w

Z =w

w − w

Re {Z }

w

2

2

2 − w

Re {Z } = − w

2

Re {Z } = w

2

Figure A.1: Probablity regions for complex modulation constellations. Z re = Re {Z } =

K max



Re {ak } =

K max

Im {ak } =

k =− K max

K max



Re {sk } p (t0 − kT )

(A.12)

K max

Im {sk } p (t0 − kT )

(A.13)

k =− K max

, Z im = Im {Z } =



k =− K max



k =− K max

, and δ ′ = w . Since (A.11) is not a tightly-bounded inequality, we can instead use 2

δ ′ = γ w to produce a tighter bound by adjusting γ > Similarly with (5.5),

1 empirically. 2

119

Pr  Z re ≥ δ ′ = 2 Pr [ Z re ≥ δ ′]

(A.14)

Pr  Z im ≥ δ ′ = 2 Pr [ Z im ≥ δ ′]

1   1 We consider a QPSK constellation sk such that Re {sk } , Im {sk } ∈  ,−  and 2  2

Re {sk } and Im {sk } are independent and uniformly distributed. Then, Pr [ Z re ≥ δ ′] = Pr [ Z im ≥ δ ′]

(A.15)

Thus (A.11) becomes 2 Pr  x(t0 , s ) ≥ w ≤ Pr  Z re ≥ δ ′ + Pr  Z im ≥ δ ′   = 2 {Pr [ Z re ≥ δ ′] + Pr [ Z im ≥ δ ′]}

(A.16)

= 4 Pr [ Z re ≥ δ ′]

From (A.12), Pr {Z re ≥ δ ′} ≤ e −νδˆ ′

K max



{

νˆ Re{ak }

E e

k =− K max

}

(A.17)

where νˆ is the solution of the following equation.

{

K max

E Re {ak } e

k =− K max

E e



νˆ Re{ak }

{

νˆ Re{ak }

}

} =δ′

(A.18)

2 Then, we can upper-bound the CCDF Pr  x(t0 , s ) ≥ w as follows.  

K max

{

}

2 νˆ Re a ) Pr  x (t0 , s ) ≥ w ≤ 4e −νδˆ ′ ∏ E e { k } ≜ Pub(QPSK , SC   k =− K max

(A.19)

120

where δ ′ = γ w . ) Similarly with the case of BPSK in section 5.1.1, we can express Pub(QPSK as follows. , SC

( BPSK ) ub , SC

P

A.4

=e

ˆ −νδ

1   2

2 K max −1

 νˆ2 p ( t0 −kT ) − νˆ2 p (t0 − kT )  +e  ∏  e k =− K max   K max

(A.20)

Modified Upper Bound for LFDMA and DFDMA

In this section, we derive the upper bound of the CCDF for LFDMA symbols which will also apply to the case of DFDMA. In chapter 3 section 3.3.2, we derived the time domain symbols after LFDMA modulation without pulse shaping as follows. We follow the symbol notations in section 3.3 of chapter 3.

xɶm = xɶQ⋅n + q

1  Q x( m )mod N , q = 0  q = j 2π  1 N −1 xp 1   ⋅ 1 − e Q  ⋅ ∑  ( n− p ) q   j 2π  +  Q    N p=0 1 − e  N Q⋅ N  

where m = Q ⋅ n + q , 0 ≤ n ≤ N − 1 , and 0 ≤ q ≤ Q − 1 . We can express Q ⋅ xɶm as follows when q ≠ 0 .

, q≠0

(A.21)

121 q j 2π  1  Q Q ⋅ xɶm = 1 − e ⋅     N

=e

=e

=e

q jπ Q



 e  

q − jπ Q

( N −1) q −Qn QN

( N −1) q −Qn jπ QN

−e

N −1

xp

∑ p=0

q jπ Q

1− e

 (n− p) q  j 2π  +  Q⋅ N   N

 1 ⋅   N

 (n− p) q  − jπ  +  Q⋅ N   N

N −1

e

p =0

 (n− p) q  − jπ  +  Q⋅ N   N

∑

e 

 q 1 ⋅ (−2 j ) sin  π  ⋅  Q N

N −1

∑ p =0

−e

xp

 (n− p ) q  jπ  +  Q⋅ N   N

e



p N

  

xp

 (Qn + q ) p (−2 j ) sin  π ⋅ −π  QN N 

 q sin π  N −1  jπ Np  Q  ⋅∑  e xp   (Qn + q ) p  p =0 N sin  π ⋅ − π   

 QN N ˆ x ≜ p    

≜ c ( n ,q , p )

=e



( N −1) q −Qn QN

N −1

⋅ ∑ c(n, q, p ) ⋅ xˆ p p =0  

(A.22)

) ≜ Z n( LFDMA ,q

 q where c(n, q, p ) = sin  π   Q

  (Qn + q ) p  − π  , xˆ p = exp ( jπ p N ) ⋅ x p , and  N sin π ⋅ QN N   

N −1

) Z n( ,LFDMA = ∑ p = 0 c(n, q, p ) ⋅ xˆ p . q

Our objective now is to characterize the following CCDF of an LFDMA signal.

122

1 M =

M −1

∑ Pr  Q ⋅ xɶ

m=0

1 M

1 = M =

1 M

≥ w 

2

m

N −1 Q −1

∑∑ Pr  Qxɶ

2

Qn + q

n = 0 q =0

≥ w 

Q −1   2  + Pr  Z ( LFDMA) 2 ≥ w  ɶ Pr Qx w ≥   Qn ∑  ∑     n,q n=0  q =1  N −1

(A.23)

Q −1   2 ) 2   Pr Pr  Z n( ,LFDMA x ≥ w + ≥ w    n ∑ ∑ q   q =1  n=0   N −1

2 Let us assume that xn has constant amplitude with unit power. Then, Pr  xn ≥ w  

becomes zero when w > 1 since xn = 1 . In such case, (A.23) becomes 2

1 M

M −1

1 2 Pr  Q ⋅ xɶm ≥ w  = ∑   M m=0

N −1 Q −1

∑∑ Pr  Z n = 0 q =1

( LFDMA ) 2 n,q

≥ w 

(A.24)

) 2 Similarly with (A.11), we bound Pr  Z n( ,LFDMA ≥ w as follows.  q  ) 2 Pr  Z n( ,LFDMA ≥ w q  

 ) ≤ Pr  Re Z n( ,LFDMA q 

{

{

= Pr  Re Z 

(

( LFDMA) n,q

{

}

2



w  ) + Pr  Im Z n( ,LFDMA q  2 

{

} ≥ δ ′ + Pr  Im {Z } ≥ δ ′ + Pr Im {Z

) = 2 Pr  Re Z n( ,LFDMA q 

( LFDMA) n,q

}

2



w 2 

} ≥ δ ′ } ≥ δ ′ )

(A.25)

( LFDMA) n ,q

1 where δ ′ = w . Or we can use δ ′ = γ w to produce a tighter bound by adjusting γ > 2 2

empirically. Using the Chernoff bound,

123

{

}

{ E {e

{

) νˆre Re Z n( LFDMA ,q

) Pr  Re Z n( ,LFDMA ≥ δ ′ ≤ e−νˆreδ ′ E e q

{

Pr  Im Z

( LFDMA ) n ,q

} ≥ δ ′ ≤ e

−νˆimδ ′

{

} } }

}

) νˆim Im Z n( LFDMA ,q

(A.26)

where νˆre and νˆim are the solutions of the following equations, respectively.

{ E {Im {Z {

}

( LFDMA ) n ,q

{

) ν Re Z n( LFDMA ,q

) E Re Z n( ,LFDMA e q

}e

{

) ν Im Z n( LFDMA ,q

} { } } − δ ′ ⋅ E {e }

{

) ν Re Z n( LFDMA ,q

−δ ′⋅ E e

{

) ν Im Z n( LFDMA ,q

}= 0 } }= 0 }

(A.27)

From (A.22) N −1

) Re Z n( ,LFDMA = ∑ c(n, q, p ) ⋅ Re { xˆ p } q

{

}

p =0

N −1   p   p = ∑ c(n, q, p ) ⋅ cos  π  Re { x p } − sin  π  Im { x p }  N p =0   N   

≜ d re ( n , q , p ) N −1

= ∑ d re (n, q, p ) p =0

N −1

{

Im Z

( LFDMA ) n,q

} = ∑ c(n, q, p) ⋅ Im {xˆ }

(A.28)

p

p=0

N −1   p   p = ∑ c(n, q, p ) ⋅ sin  π  Re { x p } + cos  π  Im { x p }  N p =0   N   

≜ dim ( n , q , p ) N −1

= ∑ dim (n, q, p ) p =0

where

d re (n, q, p ) = c(n, q, p ) ⋅  cos (π p N ) Re { x p } − sin (π p N ) Im { x p }

d im (n, q, p ) = c(n, q, p ) ⋅ sin (π p N ) Re { x p } + cos (π p N ) Im { x p } .

Using (A.28),

and

124

{

{

}

{

) ν Re Z n( LFDMA ,q

) E Re Z n( ,LFDMA e q

}

}

 N −1  N −1  = E ∑ d re (n, q, p ) exp ν ∑ d re (n, q, l )    l =0   p =0 N −1 N −1   = ∑ E d re (n, q, p )∏ eν dre ( n, q ,l )  p =0 l =0  

  N −1  ν d re ( n , q , p ) ν d re ( n , q ,l )  = ∑ E d re (n, q, p )e e  ∏ p =0 l =0   l≠ p   N −1

N −1

{

= ∑ E d re (n, q, p )eν dre ( n, q , p ) p =0

{

{

}

{

) ν Im Z n( LFDMA ,q

) E Im Z n( ,LFDMA e q N −1

{

N −1

} ∏ E {e

}

= ∑ E d im (n, q, p )eν dim ( n ,q , p ) p =0

}

ν d re ( n , q ,l )

l =0 l≠ p

N −1

} ∏ E {e

}

ν d im ( n , q ,l )

l =0 l≠ p

}

(A.29)

and

{

{

) ν Re Z n( LFDMA ,q

E e

}

} = E exp ν N −1 d (n, q, l )   = E  N −1 eν d  ∏  ∑ re  



N −1

{

}

{

}

= ∏ E eν dre ( n , q ,l )

{

{

) ν Im Z n( LFDMA ,q

E e

}

}

l =0

N −1



l =0

= ∏ E eν dim ( n , q ,l ) l =0

From (A.29) and (A.30), (A.27) becomes

 l =0

re ( n , q ,l )

  

(A.30)

125 N −1



{

E d re (n, q, p)eνˆre dre ( n ,q , p ) E e

p=0

N −1



{

νˆre d re ( n , q , p )

{

}

} =δ′

E dim (n, q, p)eνˆim dim ( n ,q , p )

{

E eνˆim dim ( n ,q , p )

p=0

}

(A.31)

} =δ′

and we can upper-bound the CCDF of an LFDMA signal as follows. 1 M

M −1

∑ Pr  Q ⋅ xɶ

m=0

1 ≤ M ≤

1 M

2

m

≥ w 

n = 0 q =1

{

N −1 Q −1



N −1

n = 0 q =1



N −1 Q −1

∑∑ 2  e ν δ E − ˆre ′

{

) νˆre Re Z n( LFDMA ,q

e

∑∑ 2  e ν δ ∏ E {eν − ˆre ′

l =0

}

}

ˆre d re ( n , q ,l )

{

}+e

−νˆimδ ′

{

) νˆ Im Z n( LFDMA ,q

+ e−νˆimδ ′ E e im N −1

∏ E {eν

}

} 

ˆim dim ( n , q ,l )

l =0

≜ Pub, LFDMA

where νˆre and νˆim are the solutions of (A.31) and δ ′ = γ w .

(A.32)

}  

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Single Carrier Orthogonal Multiple Access Technique ...

of 2007, he is with Qualcomm/Flarion Technologies, Bedminster, NJ as a senior engineer. His ... Broadband wireless mobile communications suffer from multipath ...... In 2000, a unanimous approval of the technical specifications for 3G.

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